Samir Jaber
email: jaber@chu-montpellier.fr

Audrey De Jong

Weaning from mechanical ventilation in intensive care units: a call for new international consensus guidelines

In The Lancet Respiratory Medicine, Tài Pham and colleagues1 report the results of the WEAN SAFE study, aiming to describe the epidemiology, management, timings, risk for failure, and outcomes of weaning in patients requiring at least 2 days of invasive mechanical ventilation. WEAN SAFE was an international, multicentre, prospective, observational study including 5869 critically ill adult patients, conducted in 481 intensive care units in 50 countries.1 The authors can be congratulated for this large convenience sample reporting novel and important findings. The main result was that only 3817 (65•0%) patients were successfully weaned at day 90-ie, without reintubation within 7 days of extubation. WEAN SAFE is the first study reporting data relating long-term weaning practices to outcomes from invasive mechanical ventilation in a global cohort of patients at risk for weaning failure. Pham and colleagues1 reported that higher sedation levels and increased time intervals from meeting weaning criteria to their first separation attempt were potentially modifiable factors independently associated with weaning failure. Weaning eligibility criteria were defined as follows (modified from Boles et al2): positive end-expiratory pressure less than 10 cm H 2 O, fractional concentration of oxygen in inspired air less than 0•5, not receiving paralysing agents, and receiving no or low doses of vasopressors (<0•2 μg/kg per min of norepinephrine or equivalent). It is worth noting that extubation on high-dose vasopressors, defined as more than 0•1 μg/kg per min, has been found to be a risk factor of reintubation,3 making the threshold of 0•2 μg/kg per min questionable. Moreover, the level of consciousness was excluded from the weaning criteria.2 The results might have been affected by the absence of identification of consciousness as a risk factor for delayed weaning and weaning failure. To better discriminate the influence of level of consciousness, sedation, or both, on weaning failure, Pham and colleagues1 conducted a noteworthy sensitivity analysis restricting the sample to patients without neurological impairment (post cardiac arrest, neurosurgery, or nontraumatic neurological event), excluding approximately 25% of patients. Sedation at the time of weaning readiness remained strongly associated with weaning failure.1 Randomised controlled trials have shown that reduction in sedation might improve weaning and outcomes;4-6 however, the cohort analysed by Pham and colleagues was substantially larger in terms of the number of patients enrolled, the number of participating centres, and the geographical spread than any previous weaning study. Patients were followed-up for longer timeframes than previous studies, with outcomes recorded up to day 90, and the population, invasively ventilated for more than 2 days, was at high-risk of weaning difficulties Despite well conducted statistical analyses, some limitations must be considered with the interpretation of findings. The observational design without blinded assessment of the main outcome can only infer association and not causation. All the adjustment methods are imperfect, and large, multicentre, observational studies are subject to residual confounding, such as the sickness of patients assessed by clinicians at bedside, and imprecision of the data recorded.7 Temporal biases are also likely-for example, between sedation management and outcomes, because stopping sedation will precede weaning failure-but weaning failure can also lead to prolonged sedation. Furthermore, one could say that the beginning of the weaning phase was subjectively assessed by on-site clinicians. A patient was considered to be formally in the weaning phase when a first attempt at separating a patient from the ventilator was performed, whereas the real weaning phase could have started earlier. Moreover, among extubation failures, airway failure was not differentiated from weaning failure, although risk factors of airway and weaning failure are known to be different.8 However, the limitations of the study were accurately described and no overinterpretation or inferences of causation were made.

We commend Pham and colleagues1 for exploring a complex topic with a large, multicentre, observational study. The results confirmed those from previous studies showing the higher rate of weaning failure in patients undergoing a longer period of mechanical ventilation9 as well as the negative effect of sedation and delayed weaning.6 Following the results of this study, and those by Jaber and colleagues8 and Burns and colleagues,9 a research agenda for better preventing weaning failure can be proposed (table ).10

The WEAN SAFE study confirms the results of previous studies8,9 showing great heterogeneity in the clinical practices of weaning mechanical ventilation in intensive care units and their respective effects on the outcome. This is a call for the realisation of new international consensus guidelines for weaning from mechanical ventilation after the SARS-CoV-2 pandemic and taking into account personalised medicine (ie, phenotypes of the patients in intensive care units).10


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he robustness of most systems relies on homeostasis, which itself largely depends on redundancy and feedback mechanisms. The Anthropocene, both by definition and in the idea it encapsulates, can be considered as the largest feedback system experienced by human beings: the Anthropocene concept emerged when Earth-related issues started to impact humans in a feedback loop. This essay explores the concept and implications of feedback, a step that is central to a better understanding of the Anthropocene. Researchers Enrico Costanzo and Olivier Hamant use biological analogies-a field in which concepts such as robustness, homeostasis, and feedback loops have been studied extensively. Following this viewpoint, Earth's homeostasis is comparable to that of a dynamic biological system experiencing multiple feedback loops. This naturally leads to the concept of niche construction, through which our larger habitat becomes part of our own ontology. This essay also considers frameworks that may help us to understand the interactions between humans and Earth's homeostasis, and even to question our exact relationship to planet Earth.

Defining homeostasis

The Anthropocene describes a moment in time when humanity becomes so powerful that its action becomes a threat to the balance of nature. The Anthropocene, thus, can be defined as a time when humans are changing the homeostasis of the planet.

Homeostasis (a term derived from the Greek words ὅμοιος + στάσις, meaning "the same situation") generally refers to the maintenance of stable conditions in a system. In fact, biological systems are characterized by the homeostasis of their internal parameters (e.g. concentration of biological substances, energy levels, etc.). For instance, the fact that the human body has a constant temperature of 37 °C under normal physiological conditions is well known, and any deviation from this status is a signal that "something is wrong"; everybody has experienced the discomfort of a fever.

Taking the human body as an example is indeed appropriate: the term "homeostasis" was proposed for the first time in around 1930 by Walter Cannon, a physician at Harvard, 1 although the concept may date back to the famous physician Claude Bernard, who described life as the result of the steady state of the milieu intérieur (fluid matrix). 2 Therefore, the equilibrating responses of the human body to environmental stimuli, involving feedback, are both historically and conceptually the starting point of the notion of homeostasis. 3 Based on physiology, Cannon enunciated six descriptive propositions in order to define homeostasis. 4 From biology, the concept of homeostasis has been applied in different fields, from engineering and the physical sciences to the social sciences and economics. 5

Homeostasis emerges from redundancy and complexity

What are the possible indicators of homeostasis? A property of a homeostatic system is resilience, meaning (for our purposes) the ability of the system to buffer external perturbations, and to come back to its internal state. When looking at living organisms, one can see that the number of components involved in this response can play a role in maintaining homeostasis. At an evolutionary level, natural selection is the external perturbation to which species are subjected. We can try to look to the system "biological species" as a whole, complex system, composed of individuals, which are the result of other "components" (the biological processes). What we call genetic redundancy (that is, the fact that several genes have redundant functions, i.e. are implied in producing similar-working "components," such as enzymes catalyzing the same reaction) could be seen as a buffering mechanism: the absence of a single gene usually will not lead to major defects, because otherredundantgenes or pathways will compensate for the loss. While this applies within a given generation, it also has consequences across generations and in evolution: redundancy is buffering the action of natural selection, notably by limiting the impact of mutations on essential biological functions. One may call this phenomenon "species homeostasis." Incidentally, this is probably one of the main sources of complexity in biology, explaining why biologist are unlikely to be able to "fix a radio," 6 at least in the near future.

The relation between homeostasis and complexity actually goes both ways: the multiplication of actors adds robustness to a system and thus increases its homeostasis; conversely, the system can maintain a large number of actors, including defective ones, because multiple redundant pathways are at play. Complexity promotes homeostasis, and, in turn, homeostasis allows the maintenance of complexity. Such reciprocity is also a major driving force for evolution, as defective actors in a given context can become selective advantages when external conditions change. For instance, while sickle cell anemia can be a major health problem, it also provides resistance to malaria, notably through the overproduction of an enzyme called heme oxygenase 1. 7 Therefore, with homeostasis and the maintenance of complexity comes adaptability. This also means that homogeneity and simplification, in contrast, may lead to a reduction in homeostasis, and thus more chaotic and unpredictable behaviors. A landmark of the Anthropocene is precisely this simplification. Satellite images of vast open fields illustrate how humans have switched to homogeneous crop cultivation following the precepts of an "agro-Taylorism" (see also "Imaging the Anthropocene"). Thus, either consciously or unconsciously, we have decided to loose a form of adaptability, at a larger scale than ever before. Because of this global simplification, we may have become more vulnerable to external perturbations.

Homeostasis also emerges from feedback mechanisms

While redundancy appears an effective means to reach resilience, it also has limits. In particular, the idea that a set of factors leads to similar outputs is over-simplistic. For instance, while building a high-rise, it seems reasonable that a team of engineers and architects is checking for errors. However, at each step of the construction, despite the expertise of personnel, mechanical tests are conducted to control the structural integrity of the building; in other words, the building itself contributes to the next decision during the process of its own construction. This means that in addition to redundancy, resilience also emerges from feedback systems. While in the case of constructing a building, the architects' initial decisions would dominate over the contribution made by the building in the final design, obviously this is radically different in a biological system. Biology, and homeostasis in biology, cannot be understood without taking into account such feedback. In fact, if we had to postulate the absence of feedback mechanisms in biology, this would mean that every single molecule (or group of similar molecules) has been positioned following a carefully drawn plan, and that it is blind to its environment. Because one of the main features of molecules is that they interact with one another, and thus are highly dependent on their direct environment, this view, of course, is completely wrong. Biological organisms are complex systems, and this means that they are built on the interactions between molecules, cells, and tissues. This implies that feedback occurs at every scale in this context. 8 In fact, feedback in biology does not stop at the scale of the organism, but also includes the habitat, which in the frame of the Anthropocene is probably the most relevant level.

To conclude, while redundancy allows some resilience, homeostasis primarily relies on interactions, and thus feedback systems. Because feedback is often counterintuitive and difficult to grasp (we usually better understand deterministic mechanisms and have trouble with loop reasoning), their contribution to homeostasis remains underestimated.

An origin of the concept of feedback in the Industrial Revolution

As to the Anthropocene itself, the concept of feedback probably owes much to the Industrial Revolution, and the first steam machines. One of the most cited examples is the centrifugal governor-also called the "flyball" governor, because of the two massive balls that allowed it to work-developed by, among others, James Watt. The system used very simple physical laws to self-control the amount of steam entering the machine. The "flyball" at the same time was connected mechanically both to the movement generated by the steam engine and to the valve that allowed the steam to enter the engine. Because of the steam engine movement, the "flyball" was rotating on itself. When the movement was too fast, the flyballs started to move upwards because of the increased kinetic energy, activating a system to reduce the aperture of the valve, thus allowing less steam to enter the engine, and in this way reducing the machine's speed. Eventually, the flyballs returned to their initial position until a new self-regulation occurred. 9

Later on, the concept of feedback became central in the discipline of cybernetics, founded by Norbert Wiener. 10 The question of control and the way that complex systems organize themselves goes back to the mechanisms allowing such organization, and most particularly feedback mechanisms. Selforganization is indeed a special kind of homeostasis, and this can be illustrated in the case of pattern formation. An example cited most in evolution and ecology is the Lotka-Volterra model, also called the prey-predator model. 11 In brief, a system of differential equations was adopted to explain the dynamic equilibrium between prey and predator in a given (eco)system. Such equilibrium results in a spatio-temporal "pattern" (figure 1a). a. Lotka-Volterra interactions between predators and prey: curves define the spatio-temporal variation of the number of preys and predators, showing both are strictly linked. When the amount of prey increases, the same happens for the number of predators. In contrast, when the amount of prey decreases (e.g. because of extra hunting), the same follows for the number of predators (reduced available resources).

b. Chemical activator-inhibitor system: a chemical species (A) is inhibiting another chemical species (B), which, in turn, activates the first chemical species (A). This kind of chemical interaction can result in steady-state systems originating in the so-called Turing patterns (b).

Another milestone work, surely, is the 1952 paper by Alan Turing, "The chemical basis of norphogenesis." 12 In this paper, Turing was adopting chemical reactions (diffusion reactions), described by differential equations, to explain pattern-formation in biological systems (figure 1b). 13 While these two models may seem quite different at first, the earlier being rather temporal (number of prey and predators oscillating with time) and the latter being rather spatial (generation and maintdefaultenance of distinct regional identities), they display the same fundamental pattern (figure 1).

Feedback properties: Lessons from biology

Feedback is of two kinds: either negative or positive. To illustrate the case of positive feedback, let's take the example of the pepsin protease, an enzyme that digests proteins in the stomach. This protein is secreted as an inactive form, called pepsinogen. When pepsinogen reaches the stomach lumen, a small sub-part is cleaved, and the remaining protein becomes pepsin, the active enzyme. The positive feedback lies in the step of pepsinogen cleavage: the enzyme responsible for this step is pepsin itself. In other words, pepsin synthesis requires pepsin. This mechanism ensures compartmentalization of protein degradation domains (pepsin would also digest the internal content of the cell was it not synthesized and secreted as an inactive precursor) and this example illustrates one property of positive feedback: positive feedback reinforces a starting trend. In the case of pepsinogen and pepsin, feedback ensures a rapid start to digestion by amplification of the enzymatic capacities. This is of course one of the defining points of the Anthropocene: the "Great Acceleration," a period of growth-amplification for all sectors, is typically the result of a positive feedback loop. As shown in biology, and because of their amplifying power, such positive feedback mechanisms are able to generate major changes from very subtle fluctuations. This also means that fluctuations should not be considered white noise: they can tri er positive feedback, potentially leading to complete chaos. This threat is one of the reasons why the Anthropocene attracts so much attention.

As biological systems are relatively far from behaving chaotically (for instance, similar shapes are maintained in a given species and across generations), positive feedback must be channeled in living organisms. As shown in all biological systems, negative feedback systems usually take over to stop the spirals fueled by positive feedback. For instance, the increase in body temperature brought on by fever is an effective way to increase the efficiency of enzymes, which work sub-optimally at 37 °C, and work optimally at 38 or 39 °C. Anyone who has experienced fever also knows about the sweating that accompanies it. Perspiration keeps the fever under control by cooling the body, in a negative feedback loop. Ultimately, providing the sickness is benign, this negative feedback loop will prevail and body temperature will stabilize back to 37 °C. As in biology, one of the keys to the Anthropocene is to identify such positive and negative feedback loops, in order to understand and anticipate the future trajectory of our environment, and our society.

Framing the contribution of humans in Earth homeostasis

Such simple biological feedback is so easy to grasp that even intuitive predictions can be made quite safely. However, as we now experience a "data deluge," the possible crosstalk and feedback mechanisms have reached a number so high that it becomes impossible for the Human brain to formulate any coherent picture in full confidence. 14 In fact, Human intuition is often wrong in such instances. The theory of complex systems, and its application in the form of computer modeling, has thus been developed in recent years to formalize all the possible interactions, notably including feedback, through their formalization in mathematical terms (see "Modeling Wicked Problems").

At this point, one may ask what the theoretical framework of the Anthropocene should be, and how to frame humanity in planetary homeostasis. In a time defined by climate change and global threats to ecosystems and resources, we should be able to tell the difference, say, between short-term and long-term trends, human-related ruptures of global homeostasis, and those due strictly to external factors. At the same time, such a general framework would set the basis for rational intervention, and political economic planning.

The problem of global stability was approached with particular emphasis in the second half of the twentieth century. Already before, the writing of Russian geochemist Vladimir Vernandsky had settled the concept of the biosphere, that is, the sphere of the planet Earth where life can survive, the so-called "domain of life." 15 It was not until the 1970s and 1980s, however, that a controversial theory appeared called the "Gaia hypothesis." Mainly developed by James Lovelock, 16 it is probably not by chance that the theory was formulated just as the first mass-directed green movements were rising up, mainly in response to heavy post-war industrialization in the Western world.

The Gaia hypothesis basically states that life can self-regulate the biosphere, therefore creating a planetary homeostasis suitable for sustaining life. Such homeostasis would be achieved mainly through different negative feedback mechanisms. The Gaia hypothesis stemmed from a fictional model called Daisyworld,17 in which an extremely simple biosphere was taken into account. In this model, the surface of the planet is imagined covered by a single species of "daisies." These "fictional" organisms can exist in two phenotypic forms (black or white). Here the external factor acting upon the system is radiation from the Sun. A prerequisite is the existence of an "optimal temperature" at which these "daisies" can live. In this model, "daisies" would be able to selfregulate the temperature of the planet-surface because lower radiation will favor "black-daisies" (resulting in a temperature increase), and higher radiation will favor "white-daisies" (resulting in higher albedo). Several authors have analyzed the Gaia hypothesis critically and from several points of view. 18 However, the theory aims to be coherent with natural selection, and can probably be merited with pushing scientists to investigate true life-based feedback on Earth (such as the role of forests on land or algae-produced dimethyl sulfide (DMS) in oceans). 19 Moreover, even without any relationship to the theory proposed by Lovelock, it paved the way to thinking differently about negative feedback.

Some prospects

Let's look ahead. Some practical solutions to current environmental threats and Anthropocenic challenges have already emerged. For instance, geoengineering strategies are envisioned to mitigate the greenhouse gas effects on the atmospheric temperature. In that scenario, the boundaries of human organisms, in fact, would extend to that of the planet and its atmosphere, shortcutting a large number of intermediates. Importantly, this reduction in complexity also implies that the number of potential buffering mechanisms would be reduced. Here comes the question: Is this global simplification concerning our interaction with the environment not a major tipping point in our evolution and maybe one of the most emblematic landmarks of the Anthropocene? Here, biology stresses that such simplification would be extremely harmful to the planet's homeostasis.

With the increase of the human impact on the planet, not only has our environment deteriorated, but also its dynamic has been modified dramatically. Fluctuations occur faster, meaning that feedback is less predictable and may be more difficult to channel. If the impact of humanity on the environment is starting to have a dominant effect on Earth, this may hinder the buffering effect of a more balanced complex system. This could be one of the many definitions of the Anthropocene: an era where ecosystem homeostasis is threatened by one of its dominating elements, humanity, and thus a time when the ecosystem becomes vulnerable and to some extent unpredictable, because both humanity and its environment cannot be decoupled anymore.

Beyond technology and Earth sciences, one of the most fascinating research subjects for the future is the contribution of human psychology in our relationship to Earth. In this respect, an understudied subject within the frame of the Anthropocene is the process of decision-making. In general, humans tend to take a conservative approach; when it is time to decide between two options, the conservative option, i.e. the one that is consistent with the past context, is the more likely to be picked. Such conservatism has been largely demonstrated, whether it relates to buying habits or maintaining relationships. 20 In other words, humans are not revolution-prone. In the context of the Anthropocene, such a strong bias in decision-making should be included, because it would have a major impact on the evolution of the planet, and thus on humanity indirectly.

Another field in which it is crucial to explore anthropocenic feedback mechanisms is that of education. One may expect that with better knowledge of environmental issues, humanity as a whole will decide to stay away from consumerism, which is one, if not the main, cause of the Anthropocene (also called "Capitalocene" to reflect its economic roots). Consistent with this scenario, sustainable development strategies, alternative modes of consumption and financing, are blooming everywhere: Is this enough to generate a positive feedback loop on education and to amplify this change in human civilization? Two critical points should be raised here. Sustainable development will remain an oxymoron, and thus a marketing trick, as long as development remains within the framework of unlimited economic growth. Second, the worldwide shortage of resources is fueling tensions between countries, and this, in turn, may rapidly hinder current educational efforts. In other words, are we going to change our civilization fast enough to overcome this negative feedback on education? Understanding the Earth as a complex system, where multiple feedback loops occur, means that it may also be time to consider our planet as a third party to which we are accountable. 21 An immediate consequence of this change of paradigm is that any economic transactions would be weighed up in light of what the planet can or cannot offer, including the various feedback loops, and taking into account the long-term implications. Putting a price on nature, however, is a tricky task (see "Valuing Nature"). This can be illustrated by the carbon-quota debate, and the associated risk of deadweight loss. Nevertheless, this is probably where our efforts could be focused. Re-scaling the humanenvironment nexus, investigating the many implications behind the complexity of our world, considering the psychological and temporal limits of human nature, all of these considerations might help to forecast upcoming feedback outcomes, and their surprises.

  Walter B. Cannon, "What is homeostasis?," Scientific American (January 3, 2000), online (accessed 11/06/2015); Walter B. Cannon, The Wisdom of the Body. New York: W W Norton & Co., 1932.

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Nicolas Spassky

Christian Pluta

Vesna Simic

Mohamedou Thiam

Muriel Wisniewski

STEREOCHEMICAL ASPECTS OF THE CONTROLLED RING-OPENING POLYMERIZATION OF CHIRAL CYCLIC ESTERS

The ,stereochemical aspects of controlled polymerization of lactide and pbutyrolactone are discussed. The physicochemical and thermal properties of these polymers are shown to be depending on the stereochemical structure of the macromolecular chain. Different types of processes involving change of enantiomeric composition in the course of the polymerization reaction are examined in function of different initiators used. The formation of stereocomplexes from stereocopolymers of various enantiomeric compositions is reported.

INTRODUCTION

Aliphatic polyesters are interesting materials for applications in biomedical, pharmacological and packaging applications (Ref. 1234). Synthetic polyhydroxyacids such as polylactides (PLA) and polyhydroxybutyrates (PHB) are polymers bearing asymmetric carbon atoms in the main chain . Their physicochemical and degradation properties may strongly depend on their stereochemical structure and it is possible to adjust the physical and mechanical characteristics by adjusting the balance of configurational units in the stereocopolymers . The most efficient method of synthesis of high molecular weight polyhydroxyacids is the ring-opening polymerization of the corresponding cyclic esters, i.e., lactide (LA) and P-butyrolactone (BL). The chemical synthesis of polymers with defined and controlled characteristics (structure, molecular weight) requires suitable initiators leading to a living type process. The livingness of systems in ring-opening polymerization of cyclic esters was recently discussed in terms of selectivity, defined as the ratio of rate constant of propagation reaction to the rate constant of the competing side reaction which is mainly transesterification reaction (Ref. 9). In this paper we want to discuss in the case of LA and BL monomers both of the aspects of controlled polymerizations : those related to the livingness of the systems and those dealing with stereochemical arrangements of the macromolecular chain. In addition some

0 1998 Hiithig & Wepf Verlag, Zug
CCC 1022-1360/98/$ 10.00 results on the synthesis of block copolymers and the formation of stereocomplexes between stereocopolymers of various enantiomeric compositions will be given. The presentation includes results taken from the literature and from our recent investigations mostly unpublished.

DISCUSSION AND RESULTS

Various kinds of initiators (anionic, cationic, coordination-insertion) may be used for the polymerization of lactides and lactones (Ref. 10-1 1). Aluminium alkoxide initiators (AI(OR)3, RzAIOR') were particularly studied since they lead to living type polymerization with lactides and &-caprolactone (Ref. 9,10,12,13). For BL, porphinato type aluminium initiators (Ref. 14) exhibit living type behaviour. All these initiators have a relatively low reactivity. Recently it was reported that rare-earth alkoxides show a very high reactivity in the polymerization of lactones and lactides (Ref. [15][16][17][18][19][20]. For some of them a living type behaviour was observed. High rates of polymerization were also observed in the case of anionic initiators, e.g., tBuOK,CH3C02K,CH30K with addition of crown-ethers, but some side reactions were identified (Ref. [21][22][23].

Ring-opening reaction

The ring-opening reaction in the polymerization of lactones and lactides may occur through 0-acyl or 0-alkyl cleavage. In the case of 0-acyl cleavage, the reaction proceeds with retention of configuration of the asymmetric carbon atom, while in the case of 0- alkyl ring-opening an inversion of the asymmetric center occurs and the recurring polymer unit has an opposite configuration of that of initial monomer. In the case of lactide polymerization, almost all of the initiators open the cycle through 0-acyl bound with retention of configuration. Partial racemization was reported when using some anionic initiators (Ref. 21). The situation is somewhat different in the case of BL : depending on the initiator used, 0-acyl or 0-alkyl cleavage is observed and this must be taken into account when the synthesis of a particular polyenantiomer is desired.

The initiators opening BL through 0-acyl bound include Al(OiPr)3, R2AIOR', alumoxanes, Schiff s bases Al alkoxides, R'jSnOR, distannoxanes, Zn[OAI(OR)2]2,

Z n E t r n 2 0 , ZnEt~/diol, yttrium alkoxides, while 0-alkyl cleavage is observed for CH30Na, CH3OK, tBuOK,TPPAlCl, SALCENAlCl, AlEt3/H20.

Preparation of stereocopolymers

The polymerization of mixtures of stereoisomers can proceed with no change or with a change of the initial enantiomeric composition during the reaction.

Polymerizations with no change in enantiomeric composition

In this process the enantiomeric composition of the polymer is identical to that of the starting or the reacting monomer at any stage of the reaction, i.e., at any conversion.

Three types of structural mangments in the polymer chain are possible : a) a random distribution of R,S (or RR and SS for lactide) configurational units :

The examination of triads (or higher n-ads) relations indicates that these distributions obey a Bernouillian statistics (Ref.24). A linear relation is found between optical purities of starting monomers and resulting polymers.

Typical examples of such behaviour are for example, polymerizations of lactide with Sn octoate (Ref. 25), and Zn powder (Ref.5). For BL random distributions were found for polymerizations initiated with ZnEt2/H20 (Ref. 6-7).

b) an isotactic arrangment leading to a mixture of enantiomeric -RRRRRR-or -SSSSSStype chains. In that case the process is called "stereoselective" or "enantiosymmetric" :

MON : R/S (or RR/SS) = m/n + POL : m poly R + n poly S Only partial stereoselectivity was observed in few polymerizations, for example in the case of P-BL polymerized with AIEt3/H20 initiator systems of different compositions (Ref.26,27) c) a syndiotactic arrangment of the type -RSRSRS-. Poly (P-butyro1actone)s with highly enriched alternate structures were recently prepared using some alumoxanes (Ref.28), stannoxane (Ref.7) and various alkyl tin alkoxide initiators . Tin octoate according to very recent results presents also some syndiotactic stereospecificity in polymerization of lactides (Ref. 31).

MON : R/S =m/n + POL : -RRSRSSSRRSR-

Polymerizations with change of enantiomeric composition

Two types of mechanisms can be distinguished depending on the achiral or chiral nature of the initiator.

1) GROWING CHAIN MECHANISM

In the previously studied cases when monomers with unbalanced enantiomeric compositions were polymerized in the presence of achiral initiators (Al(OiPr)g, ZnEty H20, Sn octoate, ...) no variation of optical purity (0.p.) of the unreacted monomer was noticed, i.e., no preferential consumption of one of the enantiomers occured. In our recent studies we have used for the polymerization of lactides aluminium Schiff s base alkoxide initiators. The achiral initiator derived from the simplest Schiff s base bearing the ethylene diamine moiety SALENAlOCH3 was found to be efficient for the polymerization of (D,L)-lactide in solution and a living type behaviour was observed up to 70% of conversion (Ref. 13). At higher conversions the molecular weight distribution (MWD) broadens and transesterification reactions occur. Very recently the polymerization of lactide with various enantiomeric compositions using SALENAlOCH3 initiator was performed (Ref.32). The examination of the optical properties of the prepared stereocopolymers revealed that their optical rotatory powers were higher than expected from optical purity of starting monomer feeds (Fig. 1). The microstructure of enantiomerically enriched stereocopolymers was examined by 13C NMR at high field. From the experimental values of the proportions of tetrads observed on the methine carbon atom and hexads observed on the carbonyl carbon atom, it appears clearly that the microstructure of stereocopolymers obeys a Fist order Markovian distribution of units with a reactivity ratio (rates of homo/cross propagation) equal to 2.8. This means that these stereocopolymers contain long isotactic sequences of D-and L-type units. Such type of Markovian distribution is known to fits with a growing chain mechanism (Ref.24). The properties of these lactide stereocopolymers are unusual, since whatever their enantiomeric composition all the prepared polymers are. crystalline (Fig. 2). The temperature of fusion decreases from 178OC (o.p.=lOO%) to 112OC (o.p.=40%). For stereocopolymers of lower optical purity including the racemic one, melting points around 1 5 0 T are. observed.

PO 1

MP

l8&

.. Their origin is due to the formation of stereocomplexes as subtantiated by examination of X-ray diffraction profiles. Thus the stereocomplexation may occur between chains containing relatively short enantiomeric blocks.

2) CHIRAL CATALYSIS

The second mechanism involving a change in enantiomeric composition during the reaction and requiring the presence of a chiral initiator is called "stereoelective", "enantioasymmetric" and very recently the terminology "asymmetric enantiomerdifferentiating" was proposed. The ideal definition of such a process is : "A polymerization in which starting from a mixture of enantiomeric monomer molecules, only one type of enantiomer is polymerized". This means that in the case of an exclusive choice of R-enantiomer for example, at half-reaction a pure R-type polymer is obtained and the remaining unreacted monomer is pure (S)-enantiomer. Usually the stereoelective process is not as perfect and the reaction can be expressed in the following way :

Stereoelective polymerizations were extensively studied in the case of substituted oxiranes, thiiranes and much less with p-lactones and other heterocycles .

The first case of stereoelective polymerization of (D,L)-lactide was reported by us in 1996 with a chiral aluminium alkoxide initiator SALBinaphtAlOCHg derived from a chiral Schiffs base bearing binaphthyl moiety (Ref. 36). A rather high stereoelectivity was observed. A polymer with 88% enantiomeric enrichment in D units was obtained at 19% of conversion. The polymerization reaction shows a living type character and n a n o w M W D ( R w / R n = 1.10-1.30) were obtained up to very high conversions, which indicates that transesterification reactions do not occur significantly with this sterically hindred initiator. Some results of this polymerization are reported in Table 1.

Tab. 1 Stereoelective polymerization of (D,L)-LA and (R,S)-BL using (R)SALBinaphAIOCHj as initiator system

All the obtained polymers were dextrorotatory, i.e., with predominant D-units. The stereoelectivity r = kD/kL, which is the ratio of rate constants of consumption of the two enantiomers was calculated on the basis of a first-order relation and found to be close to 20, which means that the optical purity of unreacted monomer at half reaction is around 80%. It is the highest stereoelectivity reported in asymmetric ring-opening polymerization of cyclic esters and the first example of such a high resolution in homogeneous conditions with a living type behaviour. The thermal properties prepared polylactides show again interesting features : the polymer at 19% of conversion exhibits a sharp melting peak at 145OC, a value expected for a stereocopolymer of optical purity close to 80%, while the polymer of low optical purity obtained at 90% of conversion shows a melting point at 185OC, which indicates the formation of a stereocomplex, as subtantiated by examination of X-ray diffraction pattern profiles (Ref.36).

The same SALBinaphAlOCHg initiator was used for the stereoelective polymerization of racemic P-BL. As seen from results given in Table 1, the stereoelectivity observed (r=1.3) is fifteen times lower than that observed for (D,L)lactide. The polymerization appears to have a living type behaviour since narrow MWD is found at high conversion. The magnitude of the choice (stereoelectivity) depends on the nature of guest-host association between the two partners (monomer-initiator) and is not easily predictable. We have tried to polymerize (R,S)-BL with various chiral initiators, but never the stereoelectivity exceeded 2.0 (Table 2). None of these initiators, with the exception of ZnEtY(R)-DMBD, a heterogeneous type initiator (Ref.37), which is partly isospecific, lead to crystalline polymers.

Stereocomplexes

In the previous chapter on stereocopolymers the formation of stereocomplexes was observed in the course of the polymerization of stereoisomeric mixtures of lactides with Schiff s base aluminium initiators. At the same time no formation of stereocomplexes were detected in the polymerization of BL with the same initiators. We have observed for the first time the formation of stereocomplexes during polymerization reactions in stereoselective (Ref.38) and stereoelective (Ref.39) polymerizations of racemic t- butylthiirane with zinc initiators. The presence of two enantiomeric polymeric chains of opposite configuration is required in order to obtain the formation of stereocomplexes. However the mixing of two enantiomeric polymer chains does not lead in all cases to stereocomplexes. Two types of behaviour must be distinguished. In the first case the racemic material obtained from the mixture of two polyenantiomers exhibits the same physical properties, e.g., melting point, solubility and has the same crystalline structure as the parent homopolymers. Examples of this class of polymers are : poly(methyloxirane), poly(methylthiirane), poly(hydr0xybutyrate) and some other polymers (Ref.40). Tadokoro has explained this behaviour by the presence in the racemic polymer of equal amounts of polyenantiomeric crystallites. Therefore the optically active and the racemic polymer have the same crystalline structure and the same physical properties.

In the second case the properties of the racemic blend are very different of that of parent homopolymers. For example, a raise of 50°C is observed in the melting point for the stereocomplex and the crystalline structure of the latter is different of that of the polyenantiomer. According to Tadokoro (Ref.41) the optical compensation in the material takes place in the crystalline unit cell forming a so-called racemic crystallite. As already mentionned stereocomplexes were obtained during polymerization reactions in the case of poly(t-butylthiirane) (Ref.38,42) and poly(t-butyloxirane) (Ref.43). For some other polymers the stereocomplexes were obtained by mixing both of the polyenantiomers which requires the chemical synthesis of both homopolymers by an independant way.

Stereocomplexes from blends were reported for a series of substituted poly (p-propio1actone)s (Ref.44,45) and more recently for polylactides (Ref.46,47). Differential scanning calorimetric, structural, morphological studies were reported in great detail in a series of papers by Tsuji,Ikada et a1 (Ref.48,49). The corresponding polylactides were synthesized by polymerization of D-and L-lactides using Sn octoate as initiator system. Later, Brochu, Prudhomme et a1 (Ref.50) prepared L-and D-homopolymers with aluminium isopropoxide as initiator and obtained stereocomplexes from blends. In both cases the melting points of prepared polyenantiomers were close to 180OC and those of stereocomplexes around 230-235OC (Ref. 25,50). We have prepared stereocomplexes from stereocopolymers synthesized using SALENAlOCH3 as initiator system. The nascent stereocomplex prepared from blends of pure poly L-and poly Dlactides exhibited a melting point of 247OC, somewhat higher than the previous ones. More interesting were the results obtained from blending of stereocopolymers with different optical purities. It is known that PLAS prepared using Sn octoate or Al(OiPr)3 225 (215) 210 ( 203) 198 ( 180) 187 ( 177) 213 (205) 203 (194) 193 (180) 180 (177) 205 (198) 198 (187) 187 (177) 179 (144) ".P 188 (182) 176 ( 176) 166

I as initiators lost their crystallizability when their optical purity became smaller than 76%. For PLAs prepared with Sn octoate the limit value of 0.p. for stereocomplex formation is 83% with a melting point around 18OOC (Ref. 25,49) In the case of polylactides prepared with Al(0iPr)g initiator system, the stereocomplex obtained from a polymer blend of 100USOD exhibited a melting point of 1 9 5 T (at the maximum peak) (Ref.50) As previously discussed the stereocopolymers prepared from SALENAIOCH3 initiator are crystalline whatever their optical purity. We have prepared stereocomplexes by precipitation of mixed L-and D-enriched stereocopolymers dichloromethane solutions into methanol.

Two general types of blends were studied. The first type of blends was obtained from stereocopolymers having the same optical purity, i.e., PLAX +PDAX, where X is the optical purity of the starting monomer mixture. The second type of blends was prepapred from stereocopolymers having different optical purities, i.e., PLAX + PDAY. X and Y

were vaned from 100 to 26%. The main results are presented in Table 3. All of the blends were found to form stereocomplexes and exhibited high melting points. The melting point for products of the same optical purity are much higher than those reported for polymers obtained with Sn octoate and Al(OiPr)3 initiators. The temperature of fusion is taken at the end of the melting points. Numbers in parentheses correspond to a second scan. n.p. = not prepared, but probably very similar to DA87/LA43 stereocomplex.

Polymers

It is clear that these enhanced thermal properties are again the consequence of the presence of long enantiomeric sequences in these polymers (for DA 40 or LA 40 the average length is 15 D or L units, that is 30 repeating units of the same configuration). From these results it is possible to estimate the presence of sequences containing 50 repeating units of the same configuration in the macromolecular chains of the stereocomplex obtained in the course of stereoelective polymerization of (D,L)-lactide with SALBinaphAlOCHg initiator system (Tm = 187OC) (Ref.32).

Another original route of synthesis of stereocomplexes was described in the group of Feijen (Ref.51). It consists in preparation of stereoblock copolymers by sequential ringopening polymerization of D-and L-lactides with Al(OiPr)3 initiator. The block copolymer shows a melting temperature of 205OC, which is lower than those found for stereocomplexes obtained from pure L-and D-chains, but is still much higher than that found for a pure polyenantiomer Stereocomplexes formation in the case of &-caprolactone-lactide copolymers were also reported by Murdoch and Loomis (Ref.47). We have very recently prepared diblock E- caprolactone D-or L-lactide copolymers using Y(OCHzCH20iPr)3 initiator system in dichloromethane solution at room temperature .These blocks contained around 120 &-CL units and 115 lactyl units with an average MWD = 1.3-1.4. The individual block copolymers exhibit melting points of 55OC for E-CL -sequences and around 163OC for the lactyl sequence. The blend of the two enantiomeric block copolymers produced a stereocomplex material with two melting temperatures : Tml =57OC (E-CL -sequences), Tm2 = 229°C for the lactide stereocomplex sequence. Finally another method of in situ preparation of lactide stereocomplex was recently reported by Spinu et al (Ref.54). This procedure combines polymerization with stereocomplexation by heating in the presence of Sn octoate catalyst L-LA monomer and preformed PDLA polymer. The stereocomplex is formed in situ as the polymerization proceeds. The DSC thermogram exhibits a peak at 220°C characteristic of the strereocomplex and a minor peak at 169OC corresponding to some residual PDLA

Fig. 1 .

 1 Fig. 1. Optical purities (0.p.) of prepared polylactides vs reacting monomers. Initiators : * SALENAlOCH3 (Ref. 32) * Sn Octoate (Ref 25)

Fig. 2 .

 2 Fig. 2. Melting points of lactide stereocopolymers obtained with different initiator systems : * SALENAlOCH3 (Ref 32) * Sn octoate (Ref. 25)

Tab. 2

 2 Stereoelective polymerization of (R,S)-BL in bulk using different initiators (a) Chiral moiety in the Schiffs base : Binaph =(R)-l,l-binaphtyl-2,2'diamine ; CEN = (lR,2R)-1,2 cyclohexanediamine @) using dichlorosalicylaldehyde (c) DMBD = 3,3dimethyl-l,2-butanediol (d) SEC in THF versus polystyrene standards (e) optical purity of unreacted monomer ( f ) stereoelectivity ratio (r = kR/kS) (8) by lH NMR (h) fraction insoluble in MeOH

157OC 1

 1 Temperatures of fusion of different lactide stereocomplexes prepared from different L-and D-stereocopolymers obtained with SALENAlOCH3 initiator. 138°C I 104OC I 149OC Nomenclature : LAX and DAX = polymers prepared from a lactide monomer mixture enriched in L-or D enantiomer respectively with an optical purity of X%.

  . The inter or intramolecular nature of the formed stereocomplex is not yet established. The same group have also investigated the properties of triblock lactide copolymers prepared with a central polyethylene glycol (PEG 6000) sequence. Triblocks with L-or D-lactide sequences of at least 60 lactyl units show crystallinity of the PLA phase (Tm-16OOC). The blends of triblocks with lactides of opposite configuration showed an increased melting point of the PLA phase (Tm-205-220OC) (Ref.52). More recently the same authors reported on the thermal properties of diblock copolymers of &-caprolactone and lactide enantiomers of opposite configuration. The polymers were prepared by sequential polymerization of E-caprolactone and D-or L-lactide using Y5 (F-O)(OiPr)lg as initiator system. At least 40 lactyl units are necessary to obtain the crystallization of enantiomeric lactide sequences (Tm-145-150°C). The melting points of blends of enantiomeric block copolymers is increased up to 200-210°C (Ref.53).

homopolymer. It can be mentionned that recently, on the basis of molecular simulation, a model of stereocomplex crystal growth was proposed (Ref.55).

CONCLUSIONS

The controlled living type polymerization of P-butyrolactone and lactide may be realized with a series of metal alkoxide initiators, those derived from rare-earth metals being the most reactive. The polymerization reaction of enantiomerically enriched monomers proceeds with most of the initiators with no change of the enantiomenc composition during the reaction leading to statistical type stereocopolymers. Complex aluminium alkoxide initiators derived from Schiff s bases lead on the contrary to block stereocopolymers obeing to a Markovian type distribution which are crystalline whatever the enantiomeric composition. These polymers exhibit enhanced thermal properties. Chiral initiators induce an asymmetric polymerization. A rather high stereoelectivity was observed in the case of lactide. Stereocomplexes were formed at high conversions. Lactide stereocomplexes were also prepared from blends of block stereocopolymers of various enantiomeric composition and from block copolymers, e. g . , e-caprolactone-b-lactide.


domain: spi phys

text:
Filipo Studzinski

Xavier Pucel

Jean-Loup Farges

Time is Budget: A Heuristic for Reducing the Risk of Ruin in Multi-armed Gambler Bandits

Keywords: Multi-armed bandits, Risk of ruin, Safety-critical systems

In this paper we consider Multi-Armed Gambler Bandits (MAGB), a stochastic random process in which an agent performs successive actions and either loses 1 unit from its budget after observing a failure, or earns 1 unit after a success. It constitutes a survival problem where the risk of ruin must be taken into account. The agent's initial budget evolves in time with the received rewards and must remain positive throughout the process. The contribution of this paper is the definition of an original heuristic which aims at improving the probability of survival in a MAGB by replacing the time by the budget as the factor that regulates exploration in UCB-like methods. The proposed strategy is then experimentally compared to standard algorithms presenting good results.

Introduction

Multi-Armed Bandits (MAB) constitute a classic framework to model online sequential decision-making while facing the exploration-exploitation dilemma [START_REF] Lattimore | Bandit Algorithms[END_REF][START_REF] Slivkins | Introduction to multi-armed Bandits[END_REF]. A MAB is typically represented by an agent interacting with a random process. At each successive round t, the agent chooses an action A t to perform among k possible actions and receives a corresponding reward R t . The agent must estimate the reward functions associated to each action by sampling them. Rewards resulting from a same action are independent but identically distributed, and do not give any information about other actions. In that standard version, budget and risk are not taken into account. The objective is to maximise the expected future sum of rewards [START_REF] Auer | Finite-time analysis of the multiarmed bandit problem[END_REF]. Different methods and guarantees have been proposed in the literature depending on the available information and on the assumptions on the reward distributions [START_REF] Bubeck | X-armed bandits[END_REF][START_REF] Garivier | KL-UCB-Switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints[END_REF].

Survival Multi-Armed Bandits (SMAB) [START_REF] Perotto | Open problem: risk of ruin in multiarmed Bandits[END_REF][START_REF] Riou | The survival bandit problem[END_REF] and in particular Multi-Armed Gambler Bandits (MAGB) [START_REF] Perotto | Gambler bandits and the regret of being ruined[END_REF] are recent extensions of the standard MAB problem in which the agent has a budget that must remain positive throughout the process, otherwise the agent is ruined. An initial budget B 0 = b evolves with the received rewards while the agent is alive, so as B t = B t-1 + R t if B t-1 > 0, otherwise B t = 0, the agent is ruined and the process no longer evolves. In that scenario, the agent can either increase the probability of running the process indefinitely, becoming infinitely rich, or inversely, can increase the probability of ruin, until eventually running out of budget, which means that maximising the sum of rewards requires reducing the chances of being ruined.

This paper focuses on MAGB problems [START_REF] Perotto | Gambler bandits and the regret of being ruined[END_REF], where the rewards are limited to two values, +1 and -1, and the initial budget is a positive integer. When occasionally B t = 0 is achieved for the first time, the agent is ruined. The rewards are drawn from underlying stationary Bernoulli distributions. Formally,

{k ∈ N | k ≥ 2} is the number of actions, {b ∈ N | b > 0} is the initial budget, {p i ∈ R | 0 ≤ p i ≤ 1}
is the probability of success after executing action i, which returns reward +1, and 1 -p i is the complementary probability of failure, with reward -1. It means that X t ∼ Bern(p i ) and R t = 2X t -1 for A t = i. The expected mean reward of action i is

μ i = 2p i -1.
There are few results concerning SMAB and MAGB into the literature [START_REF] Perotto | Gambler bandits and the regret of being ruined[END_REF][START_REF] Perotto | Open problem: risk of ruin in multiarmed Bandits[END_REF][START_REF] Riou | The survival bandit problem[END_REF], and the definition of an optimal algorithm is still an open problem. Related extensions like Risk Averse [START_REF] Cassel | A general approach to multi-armed bandits under risk criteria[END_REF][START_REF] Galichet | Exploration vs exploitation vs safety: riskaware multi-armed bandits[END_REF][START_REF] Sani | Risk-aversion in multi-armed bandits[END_REF][START_REF] Vakili | Risk-averse multi-armed bandit problems under mean-variance measure[END_REF], and Budgeted MAB [START_REF] Badanidiyuru | Bandits with knapsacks[END_REF][START_REF] Ding | Multi-armed bandit with budget constraint and variable costs[END_REF][START_REF] Xia | Finite budget analysis of multi-armed bandit problems[END_REF], even if sharing similar concerns, cannot be reduced to the survival setting [START_REF] Perotto | Open problem: risk of ruin in multiarmed Bandits[END_REF][START_REF] Riou | The survival bandit problem[END_REF].

Standard MAB Algorithms

Lets assume that the agent always plays each action once at the beginning of the process, so as A t = t for 1 ≤ t ≤ k, in order to provide the decision algorithm with a first observation of them. Empirical-Means is a greedy algorithm which successively chooses the action with the best estimated mean reward, A t+1 = arg max i Si,t Ni,t , where N i,t is the number of times the action i had been performed until round t, and S i,t is the sum of received rewards due to that action. That strategy is sub-optimal since no systematic exploration is performed, then it may not converge to the best action.

Exploration can be performed by introducing some non-determinism on the decision. ε-Greedy is a naive algorithm which chooses either the action with best estimated mean reward with probability ε, a hand-tuned parameter, or a random action otherwise. That strategy is sub-optimal since the exploration rate remains constant throughout the process [START_REF] Cesa-Bianchi | Finite-time regret bounds for the multiarmed Bandit problem[END_REF][START_REF] Vermorel | Multi-armed Bandit Algorithms and Empirical Evaluation[END_REF].

The standard approach for solving the exploration-exploitation dilemma is the optimism in the face of uncertainty. An intelligent exploration can be made by statistically controlling the confidence on the estimates. With similar mean reward, less explored actions should be preferred. UCB1 [START_REF] Auer | Finite-time analysis of the multiarmed bandit problem[END_REF] chooses, at each time t, the action that maximises the estimated mean plus the maximum estimation error given by a confidence bound that progressively increases over time, so as:

A t+1 = arg max 1≤i≤k S i,t N i,t + α ln(t) N i,t , ( 1
)
where α is the parameter regulating exploration. That strategy is asymptotically optimal if α is sufficiently high.

Estimating the parameters of a Bernoulli bandit corresponds to estimating the parameters of a binomial distribution. The binomial distribution represents the probability of a given number of successes on a sequence of Bernoulli trials when the parameter is known. The probability of having x successes in n trials, given

p is P(x | p, n) = Bin(p, n) = n x p x (1-p) (n-x)
. In a Bayesian approach, the beta distribution corresponds to the conjugate prior for the binomial distribution. Assuming a uniform prior, the posterior density function for p is given by f

(p | x, n) = Beta(x + 1, n -x + 1) = p x (1 -p) n-x (n + 1) n
x . Bayes-UCB [START_REF] Kaufmann | On Bayesian upper confidence bounds for bandit problems[END_REF] is an improved UCB-like method designed for Bernoulli bandits that is also asymptotically optimal. It chooses the action that maximises the 1 -1 t quantile from the Beta posterior:

A t+1 = arg max 1≤i≤k Q 1-1 /t Beta(X i,t + 1, N i,t -X i,t + 1) . ( 2
)
Finally, Thompson-Sampling is another optimal Bayesian algorithm [START_REF] Kaufmann | Thompson sampling: an asymptotically optimal finite-time analysis[END_REF]. At each round, it draws a sample from the posterior of each action to decide which one to choose. This allows a non-optimal action to be sampled with a varying frequency, which dynamically balances exploration as the posterior becomes more precise:

A t+1 = arg max 1≤i≤k [V i,t ∼ Beta(X i,t + 1, N i,t -X i,t + 1)] .
(3)

Our Contribution: The Gambler Methods

In a MAGB, the probability of being ruined by always performing action i is

1-pi pi b if p i > 1 2
, and 1 if p i ≤ 1 2 [START_REF] Perotto | Gambler bandits and the regret of being ruined[END_REF]. The expected duration of the game is b 1-2pi if p i < 1 2 , and ∞ if p i ≥ 1 2 . It means that, in a MAGB, the action with highest mean presents the best life expectancy and the best survival probability, independent of the current budget, and then, like in the classic MAB, the action with maximal mean reward is the optimal action, to which optimal methods must asymptotically converge. However, when exploring, the agent must consider the remaining budget and the estimated parameters of each action in order to compare the estimated ruin probabilities associated to them.

In this paper, we suggest a heuristic modification that can be applied to UCBlike methods, which consists in replacing t for B t into the considered equations. The intuition is that, for maximising the chances of survival, the lowest is the budget, the more the agent must favour exploitation over exploration in order to increase its budget and avoid ruin. In contrast, the higher is the budget, the more the agent should prefer a classic optimal strategy.

In this way, the Gambler-UCB method modifies UCB1 (Eq. ( 1)) by adding

α ln(Bt) Ni,t instead of α ln(t)
Ni,t to the estimated mean:

A t+1 = arg max 1≤i≤k S i,t N i,t + α ln(B t ) N i,t , (4)
and the Gambler-Bayes-UCB method modifies Bayes-UCB (Eq. ( 2)) by taking the 1 -1 /Bt quantile from the beta posterior, instead of the 1 -1 /t quantile proposed on the original method:

A t+1 = arg max 1≤i≤k Q 1-1 /B t Beta(X i,t + 1, N i,t -X i,t + 1) . ( 5

Experimental Results

The proposed algorithms, Gambler-UCB (Eq. ( 4)) and Gambler-Bayes-UCB (Eq. ( 5)) have been compared to classic and state-of-the-art MAB algorithms like UCB1 [START_REF] Auer | Finite-time analysis of the multiarmed bandit problem[END_REF], Bayes-UCB [START_REF] Kaufmann | On Bayesian upper confidence bounds for bandit problems[END_REF], and Thompson-Sampling [START_REF] Kaufmann | Thompson sampling: an asymptotically optimal finite-time analysis[END_REF], but also with naive methods like Empirical-Means and ε-Greedy [START_REF] Sutton | Introduction to Reinforcement Learning[END_REF]. The experimental scenario presents k = 10 actions, the first 8 of them parameterised by p 1 = ... = p 8 = 0.45, which corresponds to a slightly negative mean reward, leading the agent to ruin, and the two last ones defined as p 9 = 0.525 and p 10 = 0.55, meaning that both are slightly positive, but only the last one is optimal. The initial budget was set to b = k = 10.

In the experiences, the survival rate, presented in the Figure 1, corresponds to the ratio between the number of episodes running until the defined time-horizon without ruin over the total number of episodes. The proposed methods performed significantly better than the other methods considering the survival rates, corroborating the intuition. Gambler-UCB, with parameter α varying between 0.1 and 2.0, as well as Gambler-Bayes-UCB, reached survival rates around 50%. The hypothetical oracle strategy (not shown in the graphic), which always plays the best action, ensures about 80% of survival. UCB1 and ε-Greedy presented survival rates below 30%, even when the exploration parameters (α and ε, respectively) have been set to small values, losing theoretical guarantees of convergence. Bayes-UCB and Thompson-Sampling, both ensuring the best theoretical guarantees against the classic Bernoulli MAB problem, presented bad survival rates against the experienced MAGB, lower than 10%, due to intense exploration in the initial rounds. The greedy Empirical-Means method performs as well as the standard methods, reaching almost 30% of survival, corroborating the findings on [START_REF] Perotto | Gambler bandits and the regret of being ruined[END_REF].

The Figure 2 presents the average budget progression, which is affected by the survival rate. If the agent is ruined during an episode, its budget remains B t = 0 until the simulation reaches the predefined time-horizon h = 1500. In terms of budget, the performance of the proposed methods is disappointing. UCB1 and ε-Greedy with low exploration, as well as Empirical-Means, presented the best performance on the proposed setting. Even if one instance of Gambler-UCB reaches similar performance, the fact of having superior survival rates indicates that it is making sub-optimal choices too often. It means that the proposed heuristics are not converging to the optimal action, or are converging too slowly.

Conclusion and Perspectives

This paper approaches a Multi-Armed Bandit setting called MAGB, a specific survival MAB problem, in which, in addition to solve the classical explorationexploitation dilemma, the agent must find a good trade-off between safety and risk to avoid ruin, still trying to maximise the sum of rewards. Two algorithms have been proposed, Gambler-UCB and Gambler-Bayes-UCB, modifying respectively UCB1 and Bayes-UCB by replacing the time by the budget as the parameter that regulates exploration. The new methods presented good results in experimental simulations considering the survival rate, but are apparently sub-optimal in terms of convergence to the best action. Both methods are the result of a simple and intuitive heuristic, that seems to be efficient for preserving the agent alive during the initial rounds of the process, when it is more vulnerable to ruin, but the modified equations do not ensure gradative convergence to the best action. The results are nevertheless very promising, and the proposed heuristics should be the subject of theoretical analyses in future works, in order to find the necessary adjustments for ensuring optimality.

)Fig. 1 .

 1 Fig. 1. Survival rate, i.e. the proportion of episodes in which the agent reaches the time-horizon h = 1500 without ruin in n = 200 episodes.

Fig. 2 .

 2 Fig. 2. Budget progression as a function of time, averaged over n = 200 episodes.


Alessio Coi
email: alessio.coi@ifc.cnr.it

Ingeborg Barisic

Ester Garne

Anna Pierini

Marie-Claude Addor

Amaia Aizpurua Atxega

Elisa Ballardini

Paula Braz

Jennifer M Broughan

Clara Cavero-Carbonell

Hermien E K De Walle

Elizabeth S Draper

Miriam Gatt

Martin Häusler

Agnieszka Kinsner-Ovaskainen

Jennifer J Kurinczuk

Nathalie Lelong

Karen Luyt

Lorena Mezzasalma

Carmel Mullaney

Vera Nelen

Ljubica Odak

Mary T O'mahony

Isabelle Perthus

Hanitra Randrianaivo

Judith Rankin

Anke Rissmann

Florence Rouget

Bruno Schaub

David Tucker

Diana Wellesley

Katarzyna Wiśniewska

Lyubov Yevtushok

Michele Santoro

Epidemiology of aplasia cutis congenita: A population-based study in Europe

Keywords: aplasia cutis congenita, congenital anomaly, EUROCAT, population-based study 95%CI, Confidence Interval 95%

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Aplasia cutis congenita (ACC, non-syndromic (or isolated) ORPHA: 1114, OMIM: 107600) is a rare anomaly characterized by a localized or widespread absence of skin at birth. ACC mainly affects the scalp involving the midline over the skull vertex, less commonly, the underlying periosteum and bone [START_REF] Bajpai | Aplasia Cutis Cerebri With Partial Acrania-Total Reconstruction in a Severe Case and Review of the Literature[END_REF] . It usually presents as a solitary lesion, but multiple lesions can occur on the scalp or elsewhere. The individual lesion can vary from a superficial, circular or oval, well-demarcated defect or atrophic scar with alopecia to a weeping or granulating ulcer extending to the bone (non-membranous ACC). The lesions can range from a few millimetres to more than 10 cm in diameter. Some defects can have a membranous covering that can be filled with fluid, giving it a bullous appearance (membranous ACC) [START_REF]Aplasia cutis congenita[END_REF] . Although most frequently seen on the scalp, ACC can affect any part of the body, including the trunk and limbs [START_REF] Frieden | Aplasia cutis congenita: a clinical review and proposal for classification[END_REF][START_REF] Mesrati | Aplasia cutis congenita: report of 22 cases[END_REF][START_REF] Sathishkumar | Classification of aplasia cutis congenita: a 25-year review of cases presenting to a tertiary paediatric dermatology department[END_REF] . ACC can be isolated or part of a heterogeneous group of chromosomal or monogenic syndromes. Diagnosis is usually based on the findings of the clinical examination at birth. Prenatal diagnosis is limited to ACC types of genetic origin.

14683083, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jdv.18690 by Universite De Rennes 1, Wiley Online Library on [28/10/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

The aetiology of ACC remains unclear and heterogeneous. Due to the great variability of ACC expression and the associated congenital anomalies, it can be assumed that ACC results from different pathophysiological mechanisms. Numerous factors have been considered as possible causes of ACC. They include genetic factors (e.g. pathogenic variants in the BMS1 gene associated with non-syndromic ACC, in DLL4 gene associated with Adams Oliver syndrome type6, and in genes causing ectodermal dysplasia) [START_REF] Brackenrich | Aplasia Cutis Congenita[END_REF][START_REF] Chiaverini | Aplasia cutis congenita with dystrophic epidermolysis bullosa: clinical and mutational study[END_REF][START_REF] Hsu | Novel compound heterozygous ITGB4 mutations underlie lethal junctional epidermolysis bullosa with pyloric atresia and aplasia cutis congenita[END_REF][START_REF] Marneros | BMS1 Is Mutated in Aplasia Cutis Congenita[END_REF] , teratogenic agents, intrauterine infections and trauma, vascular anomalies, adhesions of the amniotic membrane to the fetal skin, amniotic rupture sequence, imperfect neural tube closure [START_REF] Drolet | Membranous Aplasia Cutis' With Hair Collars: Congenital Absence of Skin or Neuroectodermal Defect?[END_REF] . Moreover, maternal cigarette smoking has been found to be a risk factor for ACC [START_REF] Üstüner | Coexistence of aplasia cutis congenita, faun tail nevus and fetus papyraceus[END_REF] . The association between exposure to methimazole, diclofenac sodium, benzodiazepines, heparin, and valproic acid during pregnancy and ACC has also been reported [START_REF] Li | Congenital anomalies in children exposed to antithyroid drugs in-utero: a metaanalysis of cohort studies[END_REF][START_REF] Romeo | Teratogen update: Antithyroid medications[END_REF][START_REF] Martínez-Lage | Aplasia cutis congenita of the scalp[END_REF][START_REF] Sharif | Aplasia cutis congenita and low molecular weight heparin[END_REF][START_REF] Mihçi | Aplasia cutis congenita: three cases with three different underlying etiologies[END_REF][START_REF] Pajaziti | The role of diclofenac on inducing of aplasia cutis congenita: a case report[END_REF] .

Given the rarity of the condition, most information about ACC exists as individual case reports and in some medium-sized studies [START_REF] Frieden | Aplasia cutis congenita: a clinical review and proposal for classification[END_REF][START_REF] Mesrati | Aplasia cutis congenita: report of 22 cases[END_REF][START_REF] Sathishkumar | Classification of aplasia cutis congenita: a 25-year review of cases presenting to a tertiary paediatric dermatology department[END_REF][START_REF] Zhou | Systemic aplasia cutis congenita: A case report and review of the literature[END_REF][START_REF] Duan | Aplasia cutis congenita: A case report and literature review[END_REF][START_REF] Perry | Aplasia cutis congenita type V: a case report and review of the literature[END_REF][START_REF] Humphrey | A practical approach to the evaluation and treatment of an infant with aplasia cutis congenita[END_REF][START_REF] Chessa | Aplasia cutis: clinical, dermoscopic findings and management in 45 children[END_REF][START_REF] Kuemmet | Low risk of clinically important central nervous system dysraphism in a cohort study of 69 patients with isolated aplasia cutis congenita of the head[END_REF][START_REF] Schierz | Recognizable neonatal clinical features of aplasia cutis congenita[END_REF][25] .

This study aimed to investigate the epidemiology of ACC, including the prevalence of the associated anomalies, based on a large European network of population-based registries for congenital anomalies (EUROCAT).

Methods

The EUROCAT population-based congenital anomaly registries report cases diagnosed mostly up to 1 year of age with major structural congenital anomalies, chromosomal abnormalities and genetic syndromes. Live births (LB), fetal deaths (FD) with gestational age ≥ 20 weeks and terminations of pregnancy for fetal anomalies (TOPFA) following prenatal diagnosis at any gestation are registered using standardized definitions and coding 26 . Major congenital anomalies are defined in EUROCAT as structural changes that have significant medical, surgical, social or cosmetic consequences for the affected individual and typically require medical intervention.

The defined populations, the methods of case ascertainment, and the definitions and coding instructions of EUROCAT have been described in previous publications and on the EUROCAT website [26][27][START_REF]EUROCAT[END_REF][START_REF] Kinsner-Ovaskainen | A sustainable solution for the activities of the European network for surveillance of congenital anomalies: EUROCAT as part of the EU Platform on Rare Diseases Registration[END_REF][START_REF] Tucker | EUROCAT: an update on its functions and activities[END_REF] . Text descriptions reported in the database were evaluated to ensure that all the relevant clinical information was included in the study. Local registries were contacted for any additional information required. A medical Survival up to 1 week of age was estimated only for registries with a percentage of unknown/missing information less than 10%.

Proportions of ACC cases associated with EUROCAT congenital anomaly subgroups were reported. Following the EUROCAT Multiple Congenital Anomaly Algorithm, non-isolated ACC cases were further classified into: i) ACC cases associated with non-genetic and non-chromosomal multiple congenital anomalies (i.e. cases with two or more major congenital anomalies in different organ systems, where the pattern of anomalies has not been recognized as part of a genetic (single-gene) or chromosomal syndrome or sequence); ii) ACC cases associated with chromosomal anomalies; iii) ACC cases associated with genetic syndromes. The Student's t-test was used to determine eventual significant differences in birth weight, gestational age and maternal age between isolated and non-isolated live birth cases of ACC.

A p-value less than 0.05 was considered statistically significant when performing the statistical analyses.

Statistical analysis was conducted using Stata version 16 [START_REF] Statacorp | Stata Statistical Software: Release 16[END_REF] .

Results

The total number of births covered by the 28 EUROCAT registries over the 20-year study period, 1998-2017, was 10,984,537. Five hundred cases with ACC were identified during the same period, giving an overall prevalence of 5.10 per 100,000 births (95%CI: 3.83-6.80) and a live birth prevalence of 4.81 per 100,000 births (95%CI: 3.58-6.47). Out of the 500 cases collected, there were 473 live births (LB) (94.6%), nine fetal deaths (FD) (1.8%) and 18 pregnancies resulted in a termination of pregnancy for fetal anomalies (TOPFA) (3.6%).

The male to female ratio was 1.27:1.

The prevalence rates for four 5-year periods are reported in Table 1. Prevalence across the 5-year periods did not differ significantly (p = 0.798). Data on the number of cases and prevalence by registry are presented in Table 2. Prevalence between registries differed significantly (p<0.001), with the highest prevalence estimates of ACC observed for Vaud (Switzerland) and Malta registries (17.36 and 11.95 per 100,000, respectively).

[ TABLE 2] The impact of the change of ACC coding from ICD9-BPA to ICD10-BPA on prevalence estimates has been investigated including only registries (n=12) where a switch between the two systems has occurred. The Poisson regression model did not identify evidence of significant differences (p=0.169) between the prevalence estimate based on cases coded with ICD9 (4.47 per 100,000 births) and the prevalence calculated on cases coded with ICD10 (5.89 per 100,000 births).

The scalp was the most common site for ACC in our sample, affecting 96.4% (482/500) of cases. Among the 18 cases of ACC with no involvement of the scalp, extremities were the most involved (10 cases), followed by trunk (3 cases) and face and neck (2 cases). For three cases, ACC in multiple unspecified areas was reported.

We observed 331 (66.2%) isolated cases of ACC (all live births); major associated congenital anomalies were present in 169 cases (33.8% of total), 60 of which (12.0%) were ACC cases associated with chromosomal anomalies and 26 (5.2%) associated with genetic syndromes and monogenic disorders.

The proportions of ACC associated with major congenital anomalies in other organ systems are listed in Table 3. According to the EUROCAT Multiple Congenital Anomaly Algorithm 26 , associated anomalies with ACC were also categorized as multiple anomalies, genetic syndromes and chromosomal anomalies, as defined in the Methods.

[TABLE 3]

Among the associated chromosomal anomalies, Patau syndrome (trisomy 13) was the most frequent (53/60 cases, 88.3%; corresponding to 10.6% of total ACC cases) (Table 4). Four partial trisomies/monsomies of the autosomes refer to two 4q31 interstitial microdeletions, a duplication in 20q13.33 region and a 1q21.1 microdeletion syndrome.

[ TABLE 4] 14683083, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jdv.18690 by Universite De Rennes 1, Wiley Online Library on [28/10/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Among the associated genetic syndromes, Adams-Oliver was the most frequent (15/26 cases, 57.7%; corresponding to 3% of total ACC cases), followed by Johanson-Blizzard syndrome. Goltz-Gorlin syndrome, epidermolysis bullosa, Opitz G syndrome, nevus sebaceous syndrome, and Pena-Shokeir syndrome were also observed (Table 4).

Limb anomalies were the most frequent structural anomaly associated with ACC and occurred in 80 out of 500 cases of ACC (16.0%). A similar proportion was observed for ACC associated with CHD (77 out of 500 ACC cases, 15.4%). Among the 77 cases of ACC associated with CHD, 25 were severe CHD as defined in EUROCAT 26 .

The atrio-ventricular septal defects, tetralogy of Fallot, common arterial truncus, and hypoplastic left heartwere the most frequent (Table 4). In one case, ACC was associated with three severe CHD (double outlet right ventricle, hypoplastic left heart and coarctation of aorta).

Among the 36 cases of ACC associated with anomalies of the nervous system, 10 (27.8%; corresponding to 2% of total ACC cases) were associated with arhinencephaly/holoprosencephaly, followed by 7 cases each of neural tube defects, agenesis of corpus callosum, and severe microcephaly; ACC associated with other congenital anomalies were often part of a chromosomal syndrome. This was the case for 88.5% (23/26) of the congenital anomalies of the eye, 66.7% (10/15) of the ear, face and neck, 61.1% (22/36) of the nervous system, 63.6% (7/11) of the oro-facial clefts and 50.6% (39/77) of CHDs.

Concerning the association with possible known teratogens, five mothers took benzodiazepines in the first trimester of pregnancy (from the 1 st day of the last menstrual period to the 12 th week of gestation), two mothers took methimazole or heparin. None of the mothers took diclofenac or valproic acid. Among live births, differences in birth weight, gestational age, and maternal age between isolated ACC and associated with major congenital anomalies ACC are reported in Table 5.

[ TABLE 5] Statistically significant differences (p<0.05) were observed between isolated and non-isolated ACC in birth weight for term-born infants (gestational age ≥37 weeks), especially for ACC associated with genetic syndromes (2831 grams vs 3697 grams). On the contrary, no differences in gestational age and maternal age were found between isolated and non-isolated ACC.

Among the 452 cases (out of 500) with available information about survival beyond one week of age, 18 (4.0%) died within the first week. As expected, most deaths were observed for ACC associated with chromosomal anomalies (15/18, 83.3%). No deaths before one week of age were reported for liveborn infants with isolated ACC.

A family history of ACC was found in 10 patients (10/500, 2.0%): five with ACC from the father's family, four from the mother's family (one maternal relative with Adams-Oliver syndrome) and one from both mother's and father's families with ACC. more complex cases with associated congenital anomalies referred for treatment at a specialised centre, whereas in our population-based study more isolated milder cases are identified, representing the reality of a wider range of the true pathophysiology.

We observed 16% of cases associated with limb anomalies (Type 2 ACC, according to Frieden's classification 3 ), that is almost in line with what was found by Sathishkumar et al. [START_REF] Sathishkumar | Classification of aplasia cutis congenita: a 25-year review of cases presenting to a tertiary paediatric dermatology department[END_REF] (12.9%) but higher than in Yang et al. 25 and Mesrati et al. [START_REF] Mesrati | Aplasia cutis congenita: report of 22 cases[END_REF] (5.1% and 9.1%, respectively).

ACC is often included as part of the Adams-Oliver syndrome, a rare disorder characterized by the combination of congenital limb anomalies and scalp defects, often accompanied by skull ossification defects.

We found that 3.2% of our ACC cases had Adams-Oliver syndrome, a lower proportion than Mesrati et al. [START_REF] Mesrati | Aplasia cutis congenita: report of 22 cases[END_REF] , reporting 9% of Adams-Oliver syndrome cases in their series of 22 cases of ACC.

We observed three cases (0.6%) associated with epidermolysis bullosa, corresponding to Frieden's Type 6.

This association was also observed by Yang et al. 25 (1 out of 59, 1.7%) and Schierz et al. [START_REF] Schierz | Recognizable neonatal clinical features of aplasia cutis congenita[END_REF] (3 out of 37, 8.1%).

Comparing isolated and non-isolated ACC among live birth cases, we found a significantly lower birth weight in term-born infants with ACC associated with genetic syndromes than in term-born infants with isolated ACC. This is also well known for other anomalies and similar differences in birth weight were also observed by Schierz et al. [START_REF] Schierz | Recognizable neonatal clinical features of aplasia cutis congenita[END_REF] .

This study has some limitations; in particular, potential variation due to coding practices, completeness of data sources, and case description accuracy must be considered when combining epidemiological data from many registries. Moreover, clinical data might not be as detailed as in a case report or in hospital-based case series; for example, no information was collected in EUROCAT regarding associated fetus papyraceus or placental infarct, thus Frieden's Type 5 ACC cases cannot be identified.

Conclusion

Epidemiology studies on rare congenital anomalies are usually difficult because sufficiently large standardized cohorts are difficult to obtain. In this study we presented the first epidemiological populationbased study on ACC, since most of the published studies are case reports or medium-sized hospital-based retrospective studies. Our study is based on a large European population and a standardized data collection, this providing great statistical power to the epidemiological study on such a rare congenital anomaly compared to hospital-based case series. In particular, the methodology of data collection exploited within the EUROCAT guarantee the production of reliable estimates both in terms of prevalence for all birth outcomes and proportion of associated anomalies.

  Twenty-eight EUROCAT full member registries in 16 European countries participated in the study. All cases of ACC born between January 1, 1998 and December 31, 2017, coded with the International Classification of Diseases, ninth (ICD-9) or tenth revision (ICD-10) with British Paediatric Association (BPA) one-digit extensions for ACC (ICD9-BPA, 75739; ICD10-BPA, Q8480) were extracted from the EUROCAT Central Database which is operated by the JRC-EUROCAT Central Registry, European Commission Joint Research Centre in Ispra (Italy). Registries submit individual anonymized records of cases of congenital anomalies; thus, no ethical approval for the study was required.

  geneticist (IB) and a paediatrician (EG) reviewed all anomaly codes and written text descriptions of the anomalies in this study.The overall and live birth prevalence were estimated using Poisson regression with random effects models allowing the prevalence in different registries to vary. 95% confidence intervals (95%CIs), based on the Poisson distribution, were calculated for prevalence estimates. The χ2 test for homogeneity was performed to assess whether differences in prevalence estimates across registries reflected actual differences or were due to random fluctuation. A Poisson regression model was also used to test time trends in prevalence between the four 5-year periods 1998-2002, 2003-2007, 2008-2012, and 2013-2017 (reference baseline period: 1998-2002). An analogous model was used to evaluate the impact of the change of coding for ACC from the unspecific ICD9-BPA code 75739 (Other specified anomalies of skin) and the specific ICD10 code Q8480.

  14683083, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jdv.18690 by Universite De Rennes 1, Wiley Online Library on [28/10/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

TABLE 1 ]

 1 14683083, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jdv.18690 by Universite De Rennes 1, Wiley Online Library on [28/10/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License[

Table 2 .

 2 Number of cases and prevalence (per 100,000 births) of aplasia cutis congenita in 28 EUROCAT registries.

	Registry	Years included Total births Number	Prevalence (95%CI)
				of cases
	Vaud (Switzerland)	1998-2017	155,522	27	17.36 (11.44-25.26)
	Malta	1998-2017	83,668	10	11.95 (5.73-21.98)
	Tuscany (Italy)	1998-2017	566,324	56	9.89 (7.47-12.84)
	Odense (Denmark)	2000-2015	81,392	8	9.83 (4.24-19.37)
	Brittany (France)	2011-2017	244,690	24	9.81 (6.28-14.59)
	Basque Country (Spain)	1998-2016	367,440	32	8.71 (5.96-12.29)
	Auvergne (France)	1998-2017	268,338	23	8.57 (5.43-12.86)
	Wielkopolska (Poland)	1998-2017	741,725	55	7.42 (5.59-9.65)
	Saxony Anhalt (Germany)	1998-2017	336,187	23	6.84 (4.34-10.27)
	Cork & Kerry (Ireland)	1998-2017	181,756	11	6.05 (3.02-10.83)
	Isle de Reunion (France)	2002-2017	232,043	13	5.60 (2.98-9.58)
	Wales (UK)	1998-2017	668,205	37	5.54 (3.90-7.63)
	Antwerp (Belgium)	1998-2016	370,959	20	5.39 (3.29-8.33)
	Emilia Romagna (Italy)	1998-2017	690,381	35	5.07 (3.53-7.05)
	Styria (Austria)	1998-2016	199,998	7	3.50 (1.41-7.21)
	Paris (France)	1998-2017	561,416	18	3.21 (1.90-5.07)
	North Netherland	1998-2017	360,762	10	2.77 (1.33-5.10)
	OMNI-Net (Ukraine)	2005-2016	360,948	10	2.77 (1.33-5.10)
	East Midlands & South Yorkshire (UK)	1998-2012 & 2016-2017	1,143,462	30	2.62 (1.77-3.75)
	South East Ireland	1998-2017	137,175	3	2.19 (0.45-6.39)
	South Portugal	1998-2017	366,939	8	2.18 (0.94-4.30)
	Northern England (UK)	2000-2017	575,155	12	2.09 (1.08-3.64)
	South West England (UK)	2005-2017	641,971	12	1.87 (0.97-3.27)
	Valencian Region (Spain)	2007-2016	489,361	8	1.63 (0.70-3.22)
	French West Indies (France)	2009-2017	85,250	1	1.17 (0.03-6.54)
	Thames Valley (UK)	2005-2017	387,555	4	1.03 (0.28-2.64)
	Wessex (UK)	1998-2017	570,130	3	0.53 (0.11-1.54)
	Zagreb (Croatia)	1998-2015	115,785	0	0.00
	TOTAL		10,984,537	500	5.10 (3.83-6.80)

Table 3 .

 3 Proportions of aplasia cutis congenita associated with EUROCAT congenital anomaly subgroups. Associated cases are also reported according to the EUROCAT Multiple Congenital Anomaly Algorithm (multiple anomalies, genetic syndromes, and chromosomal anomalies). The sum of the percentages does not equal to 100% because each aplasia cutis congenita case can be associated with more than one anomaly across different subgroups. Proportions are calculated on 500 total aplasia cutis congenita cases. § Cases with two or more major congenital anomalies in different organ systems, where the pattern of anomalies has not been recognized as part of a monogenic or chromosomal syndrome or sequence. CHD = Congenital heart defects.

	Associated anomaly	Total *	Multiple anomalies §	Genetic syndromes	Chromosomal anomalies
	subgroups	n (%)	n (%)	n (%)	n (%)
	Limb	80 (16.0)	29 (36.2)	14 (17.5)	37 (46.3)
	CHD	77 (15.4)	29 (37.7)	9 (11.7)	39 (50.6)
	Nervous system	36 (7.2)	10 (27.8)	4 (11.1)	22 (61.1)
	Eye	26 (5.2)	3 (11.5)	0 (0.0)	23 (88.5)
	Genital system	26 (5.2)	12 (46.2)	3 (11.5)	11 (42.3)
	Urinary system	25 (5.0)	11 (44.0)	4 (16.0)	10 (40.0)
	Digestive system	23 (4.6)	9 (39.1)	6 (26.1)	8 (34.8)
	Oro-facial cleft	11 (2.2)	2 (18.2)	2 (18.2)	7 (63.6)
	Ear, face and neck	15 (3.0)	3 (20.0)	2 (13.3)	10 (66.7)
	Respiratory system	6 (1.2)	2 (33.3)	3 (50.0)	1 (16.7)
	*

Table 4 .

 4 Distribution of associated congenital anomalies among the groups of chromosomal anomalies, genetic syndromes, congenital heart defects (CHD) and anomalies of the nervous system. The sum of the percentages of each group does not equal to 100% because some aplasia cutis congenita cases can be associated with more than one anomaly.

	Group	n	%	% on total ACC cases (n=500)
	Chromosomal anomalies (n=60)
	Patau syndrome (Trisomy 13)	53 88.3	10.6
	Partial trisomies/monosomies of the autosomes 4	6.7	0.8
	Klinefelter syndrome	1	1.7	0.2
	Deletion of X chromosome	1	1.7	0.2
	Karyotype 47, XYY	1	1.7	0.2
	Genetic syndromes (n=26)
	Adams-Oliver syndrome	15 57.7	3.0
	Johanson-Blizzard syndrome	4	15.4	0.8
	Goltz-Gorlin syndrome	2	7.7	0.4
	Epidermolysis bullosa	2	7.7	0.4
	Opitz G syndrome	1	3.8	0.2
	Nevus sebaceous syndrome	1	3.8	0.2
	Pena-Shokeir syndrome	1	3.8	0.2
	Severe CHD (n=25)
	Atrio-ventricular septal defects	5	6.5	1.0
	Hypoplastic left heart	5	6.5	1.0
	Tetralogy of Fallot	4	5.2	0.8
	Common arterial truncus	4	5.2	0.8
	Transposition of great arteries	2	2.6	0.4
	Pulmonary valve atresia	2	2.6	0.4
	Coarctation of the aorta	2	2.6	0.4
	Hypoplastic right heart	1	1.3	0.2
	Aortic valve atresia/stenosis	1	1.3	0.2
	Double outlet right ventricle	1	1.3	0.2
	Anomalies of the nervous system (n=36)
	Arhinencephaly/holoprosencephaly	10 27.8	2.0
	Neural tube defects	7	19.4	1.4
	Agenesis of corpus callosum	7	19.4	1.4
	Severe microcephaly	7	19.4	1.4
	Hydrocephalus	2	5.6	0.4
	Cerebral cysts	2	5.6	0.4
	Malformation of brain	2	5.6	0.4
	Arnold-Chiari syndrome	1	2.8	0.2

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Acknowledgements 14683083, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jdv.18690 by Universite De Rennes 1, Wiley Online Library on [28/10/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

We thank JRC-EUROCAT Central Registry, European Commission, Joint Research Centre (JRC), Ispra, Italy, for the data management and selection of cases included in the study. We also thank many people throughout Europe involved in providing and processing information, including affected families, clinicians, health professionals, medical record clerks, and registry staff.

There was no specific funding for this study.

valid requests and with permission of the participating registries of congenital anomalies.

14683083, ja, Downloaded from https://onlinelibrary.wiley.com/

Discussion

This study, using data from EUROCAT, a large European network of population-based registries for surveillance of congenital anomalies, represents the most extensive series of cases with ACC in Europe ever published.

To our knowledge, this is the first population-based study on ACC. Most of the studies on ACC are case reports and single hospital-based series mainly reporting clinical features. Whereas, our study is not based on disease-specific registries; instead, it is based on European population-based registries that register a wide range of congenital anomalies. For this reason, the different design of our study must be considered when comparing our results to case series and hospital series.

In this population-based study, the prevalence of ACC from 1998 to 2017 was 5.10 per 100,000 births, that is in line with the Orphanet estimate 32 . The prevalence estimate remained stable across the four 5-year periods, but with significant heterogeneity across the European registries. Differences in case ascertainment methods, especially for mild forms of ACC, may contribute to the observed geographical differences, but true regional differences cannot be excluded. To limit the effect of such high variability, we used a meta-analytical approach to produce an overall prevalence estimate of ACC.

The change from the unspecific ICD9-BPA code to the specific ICD10-BPA code did not significantly impact the prevalence (4.47 vs 5.89 per 100,000, respectively). This suggests that both the EUROCAT method of registration of cases (using ICD9-BPA in combination with a written text description) and the standardized inclusion/exclusion criteria were well established even in the early years and before the introduction of a specific ICD10-BPA code for ACC.

The male-to-female ratio was greater than 1 in our study, in line with what is commonly reported in the literature [START_REF] Sathishkumar | Classification of aplasia cutis congenita: a 25-year review of cases presenting to a tertiary paediatric dermatology department[END_REF]25 . We found that the scalp was the most common site for ACC and more frequent (96.4%) than reported by Frieden (86.0%) [START_REF] Frieden | Aplasia cutis congenita: a clinical review and proposal for classification[END_REF] , Sathishkumar et al. [START_REF] Sathishkumar | Classification of aplasia cutis congenita: a 25-year review of cases presenting to a tertiary paediatric dermatology department[END_REF] (82.1%), and Mesrati et al. [START_REF] Mesrati | Aplasia cutis congenita: report of 22 cases[END_REF] (72.7%).

We observed that 66% of cases presented isolated ACC, which can be treated differently depending on the defect's extent. Using conservative wound care, healing by secondary intention is preferred when possible 33 . Larger lesions (3-4 cm) may require bone or skin grafts. After healing, scar revision can be considered.

However, from a clinical point of view, it is essential after a diagnosis of ACC to assess the eventual presence of other and more severe associated anomalies that may affect the prognosis. In this case, additional treatments may be required.

Concerning the associated anomalies observed in our cohort, a comparison could be made with the paper of Schierz et al. [START_REF] Schierz | Recognizable neonatal clinical features of aplasia cutis congenita[END_REF] , who conducted a retrospective hospital-based study on 37 newborns, comparing isolated and non-isolated forms of ACC. The authors declared 49% of the isolated form of ACC. We found a higher proportion of isolated ACC than Schierz et al. [START_REF] Schierz | Recognizable neonatal clinical features of aplasia cutis congenita[END_REF]