Loubna Mazini

Nourredine Matar

Said Bouhya

Diaa Marzouk

Wagida Anwar

Meriem Khyatti

Mazini L1

Matar N2#

Bouhya S3#, Marzouk

Umbilical Cord Blood Banking for Transplantation in Morocco: Problems and opportunities

Keywords: Umbilical Cord Blood Banking, Related donor, Unrelated donor, Legislation, Morocco 1

INTRODUCTION

Umbilical Cord Blood transplantation (UCBT) was first reported in 1989 by Gluckman E. et al [START_REF] Gluckman | Hematopoietic reconstitution in a patient with Fanconi's anemia by means of umbilical cord blood from an HLA-identical sibling[END_REF] and UCB is now routinely used as a source of Heamatopoietic Stem Cells (HSC). The major advantage of this source is the acceptable degree of human leukocyte antigen (HLA) mismatch compared to bone marrow (BM) and is more and immediately accessible when no available BM HLA-matched donors are found [START_REF] Rocha | Comparison of outcomes of unrelated bone marrow and umbilical cord blood transplants in children with acute leukemia[END_REF][START_REF] Dalle | Results of an unrelated transplant search strategy using partially HLAmismatched cord blood as an immediate alternative to HLA-matched bone marrow[END_REF][START_REF] Barker | Transplantation of 2 partially HLA-matched umbilical cord blood units to enhance engraftment in adults with hematologic malignancy[END_REF][START_REF] Eapen | Effect of graft source on unrelated donor haematopoietic stem-cell transplantation in adults with acute leukemia: a rretrospective analysis[END_REF]. In addition, UCB presents less risks of transmissible viral infection than maternal blood [START_REF] Kogler | Hematopoietic transplant potential of unrelated cord blood : Critical issues[END_REF], and stem cells it contains present self-renewal, proliferative and immune-modulatory abilities [START_REF] Eapen | Outcomes of transplantation of unrelated donor umbilical cord blood and bone marrow in children with acute leukemia: a comparison study[END_REF]. Furthermore, although UCB contains one log less nucleated cells than BM and Peripheral Blood (PB), UCB units are now successfully used in transplantation of patients with myeloid and lymphatic leukemia, lymphoma, myelodysplasia, aplastic anaemia, heamoglobinopathies, thalassaemia, immune deficiency, autoimmune and inherited metabolic diseases, and other diseases especially in children [START_REF] Gluckman | Hematopoietic reconstitution in a patient with Fanconi's anemia by means of umbilical cord blood from an HLA-identical sibling[END_REF][START_REF] Rocha | Comparison of outcomes of unrelated bone marrow and umbilical cord blood transplants in children with acute leukemia[END_REF][START_REF] Dalle | Results of an unrelated transplant search strategy using partially HLAmismatched cord blood as an immediate alternative to HLA-matched bone marrow[END_REF][START_REF] Barker | Transplantation of 2 partially HLA-matched umbilical cord blood units to enhance engraftment in adults with hematologic malignancy[END_REF][START_REF] Eapen | Effect of graft source on unrelated donor haematopoietic stem-cell transplantation in adults with acute leukemia: a rretrospective analysis[END_REF][START_REF] Kogler | Hematopoietic transplant potential of unrelated cord blood : Critical issues[END_REF][START_REF] Eapen | Outcomes of transplantation of unrelated donor umbilical cord blood and bone marrow in children with acute leukemia: a comparison study[END_REF][START_REF] Wagner | Transplantation of unrelated donor umbilical cord blood in 102 patients with malignant and nonmalignant diseases: influence of CD34 cell dose and HLA disparity on treatment-related mortality and survival[END_REF]. However, allogeneic UCBT is limited by HLA-matching and many previous reports correlated outcomes in engraftments to HLA disparity after UCBT in patients [START_REF] Dalle | Results of an unrelated transplant search strategy using partially HLAmismatched cord blood as an immediate alternative to HLA-matched bone marrow[END_REF][START_REF] Barker | Transplantation of 2 partially HLA-matched umbilical cord blood units to enhance engraftment in adults with hematologic malignancy[END_REF][START_REF] Wagner | Transplantation of unrelated donor umbilical cord blood in 102 patients with malignant and nonmalignant diseases: influence of CD34 cell dose and HLA disparity on treatment-related mortality and survival[END_REF][START_REF] Atsuta | HLA Working Group of the Japan Society for Hematopoietic Cell Transplantation. Different effects of HLA disparity on transplant outcomes after single-unit cord blood transplantation between pediatric and adult patients with leukemia[END_REF].

Many other variables are influencing outcomes after UCBT in adults such as diagnosis and stage of disease, but the nucleated cell dose has been reported to be the clearly associated with the engraftment and the risk of transplant-related events [START_REF] Migliaccio | Cell dose and speed of engraftment in placental/umbilical cord blood transplantation : graft progenitor cell content is a better predictor than nucleated cell quantity[END_REF]. To overcome this cell dose restriction, double or multiple units of UCB are used in children and adolescent [START_REF] Kang | Double umbilical cord blood transplantation for children and adolescents[END_REF] and in adults with a reduced intensity conditioning to benefit older patients or those for which myeloablative conditioning remains too risky [START_REF] Barker | Transplantation of 2 partially HLA-matched umbilical cord blood units to enhance engraftment in adults with hematologic malignancy[END_REF][START_REF] Ballen | Double unrelated reduced-intensity umbilical cord blood transplantation in adults[END_REF][START_REF] Brunstein | Umbilical cord blood transplantation after non-myeloablative conditioning: impact on transplantation outcomes in 110 adults with hematologic disease[END_REF][START_REF] Brunstein | Allogeneic hematopoietic cell transplantation for hematologic malignancy: relative risks and benefits of double umbilical cord blood[END_REF].

As the immunologic complications of Graft Versus Host Disease (GVHD) usually offset the therapeutic benefits of unrelated BM transplants, the UCB selection criteria for unrelated transplantation were different regarding malignant and non-malignant diseases [START_REF] Kang | Double umbilical cord blood transplantation for children and adolescents[END_REF], and specific strategies and transplant programs are improved to better help finding of adequate grafts quality.

With the increasing therapeutic applications of related and unrelated UCB stem cells followed by the increased demand of UCB units, international organizations, registries and biobanks are created to improve good practices in UCB collection, manipulation and storage to achieve high quality for the UCB units, and to define standardized procedures for donor search and acquisition.

(http://factwebsite.org/CordSearch.aspx?&type=CordBloodBank&country=&state=)

FOCUS ON CORD BLOOD TRANSPLANTATION

Hematology disorders/ Oncology

Several studies have shown that patients with acute leukemia are the most treated with UCB units in the absence of Matched-Related Donor (MRD) or Unrelated Donor (URD). After UCB HSC Transplantation (HSCT), the incidence and severity of GVHD is less than observed with BM and PB [START_REF] Gluckman | Hematopoietic reconstitution in a patient with Fanconi's anemia by means of umbilical cord blood from an HLA-identical sibling[END_REF][START_REF] Rocha | Comparison of outcomes of unrelated bone marrow and umbilical cord blood transplants in children with acute leukemia[END_REF][START_REF] Kang | Double umbilical cord blood transplantation for children and adolescents[END_REF]. This may be due to the functional immaturity of infused lymphocytes with a decreased cytotoxicity, an altered cytokine profile, a decreased HLA expression and an increased regulatory T cells [START_REF] Takahashi | Comparative signle-institute analysis of cord blood transplantation from unrelated donors with bone marrow or peripheral blood stem-cell transplants from related donors in adult patients with hematologic malignancies after myeloablative conditioning regimen[END_REF][START_REF] Rocha | Graft-versus-host-disease in children who have received a cord-blood or bone marrow transplant from an HLA-identical sibling. Eurocord and International Bone Marrow Registry Working Committee on Alternative Donor and Stem cell Sources[END_REF][START_REF] Szabolcs | Immune reconstitution after unrelated cord blood transplantation[END_REF]. Other findings suggest that even if survival is similar after transplantation with MRD, URD and UCB in children with acute leukemia, occurrence of acute and chronic GVHD and relapse remain higher respectively in MRD and URD than in UCB group [START_REF] Yi | Hematopoietic stem cell transplantation in children with acute leukemia: similar outcomes in recipients of umbilical cord blood versus marrow or peripheral blood stem cells from related or unrelated donors[END_REF], while survival after transplantation with URD UCB units is similar to Matched URD from BM [START_REF] Barker | Survival after transplantation of unrelated donor umbilical cord blood is comparable to that of human leukocyte antigen-matched unrelated donor bone marrow: results of a matched-pair analysis[END_REF].

Furthermore, the outcomes of related or unrelated UCBT in children revealed the importance of the cell dose infused as a predictor of neutrophils and platelets engraftment [START_REF] Wagner | Transplantation of unrelated donor umbilical cord blood in 102 patients with malignant and nonmalignant diseases: influence of CD34 cell dose and HLA disparity on treatment-related mortality and survival[END_REF][START_REF] Gluckman | Outcome of cord-blood transplantation from related and unrelated donors. Eurocord Transplant Group and the European Blood and Marrow Transplantation Group[END_REF]. In addition, the incidence of GVHD and chronic GVHD depends on the degree of HLA mismatch in both pediatric [START_REF] Rocha | Comparison of outcomes of unrelated bone marrow and umbilical cord blood transplants in children with acute leukemia[END_REF][START_REF] Dalle | Results of an unrelated transplant search strategy using partially HLAmismatched cord blood as an immediate alternative to HLA-matched bone marrow[END_REF][START_REF] Rocha | Graft-versus-host-disease in children who have received a cord-blood or bone marrow transplant from an HLA-identical sibling. Eurocord and International Bone Marrow Registry Working Committee on Alternative Donor and Stem cell Sources[END_REF][START_REF] Yi | Hematopoietic stem cell transplantation in children with acute leukemia: similar outcomes in recipients of umbilical cord blood versus marrow or peripheral blood stem cells from related or unrelated donors[END_REF][START_REF] Wagner | Successful transplantation of HLA-matched and HLA-mismatched umbilical cord blood from unrelated donors : analysis of engraftment and acute graft-versus-host disease[END_REF][START_REF] Kutzberg | Results of the Cord Blood Transplantation Study (COBLT): clinical outcomes of unrelated donor umbilical cord blood transplantation in pediatric patients with hematologic malignancies[END_REF][START_REF] Smith | Hematopoietic cell transplantation for children with acute lymhpoblastic leukemia in second complete remission: similar outcomes in recipients of unrelated marrow and umbilical cord blood versus marrow from HLA-matched sibling donors[END_REF][START_REF] Yoo | Current status of pediatric umbilical cord blood transplantation in Korea: a multicenter retrospective analysis of 236 cases[END_REF][START_REF] Kato | Cord blood transplantation from unrelated donors for children with acute lymphoblastic leukemia in Japan : the impact of methotrexate on clinical outcomes[END_REF] and adult patients [START_REF] Barker | Transplantation of 2 partially HLA-matched umbilical cord blood units to enhance engraftment in adults with hematologic malignancy[END_REF][START_REF] Kang | Double umbilical cord blood transplantation for children and adolescents[END_REF][START_REF] Takahashi | Comparative signle-institute analysis of cord blood transplantation from unrelated donors with bone marrow or peripheral blood stem-cell transplants from related donors in adult patients with hematologic malignancies after myeloablative conditioning regimen[END_REF][START_REF] Fernandes | Enrocord and Inborn Errors working party of european group for blood ans marrow transplantation. Transplantation in patients with SCID: mismatched related stem cells or unrelated cord blood?[END_REF][START_REF] Ballen | Double unrelated reduced-intensity umbilical cord blood transplantation in adults[END_REF][START_REF] Brunstein | Umbilical cord blood transplantation after non-myeloablative conditioning: impact on transplantation outcomes in 110 adults with hematologic disease[END_REF][START_REF] Rocha | Transplants of umbilical-cord blood or bone marrow from unrelated donors in adults with acute leukemia[END_REF][START_REF] Laughlin | Outcomes after transplantation of cord blood or bone marrow from unrelated donors in adults with leukemia[END_REF][START_REF] Kumar | Allogeneic hematopoietic stem cell transplantation in adult acute lymphocytic leukemia: impact of donor source on survival[END_REF] as indicated in Table 1 and2. In fact, the infusion of partially HLA-matched UCB units, neutrophils and platelets engraftment ranged from 26-32 days (Table 1) and 20-27 days (Table 2) in pediatric and adult patients respectively. The overall survival also increased in groups with rapid neutrophils engraftment.

Non oncology

Most indicated use of UCB units in non heamatopoietic diseases remains the primary immune deficiencies. HSCT from an HLA-matched donor was in fact shown to correct the immune system in children [START_REF] Frangoul | Umbilical cord blood transplantation in children with immune deficiency : results of a multicenter study[END_REF]. As indicated in Table 3, successful and over survival after UCBT was found to be dependent on the degree of HLA disparity [START_REF] Bizzetto | Outcomes after related and unrelated umbilical cord blood transplantation for heredity bone marrow failure syndromes other than Fanconi anemia[END_REF]. Other studies suggested that higher infused cell doses can partially reduce the impact of HLA disparity on survival [START_REF] Wagner | Transplantation of unrelated donor umbilical cord blood in 102 patients with malignant and nonmalignant diseases: influence of CD34 cell dose and HLA disparity on treatment-related mortality and survival[END_REF]. Public banks are for non-profit and financed by public funds. They are involved in collecting from anonymous consent donors after baby's delivery, processing and conserving UCB units in a registry for allogenic use. For inclusion in the registry, samples are screened based on volume, cell number, tissue types, health history and infectious diseases status.

The units are then made available to suitably matched recipients. A great societal benefit is obtained from this UCB altruist donation especially for populations with ethnic minorities. Registries of these units in public banks has became a mandatory instrumental tool for heamatopoietic transplantation or applications in regenerative medicine. Reports from the Bone Marrow Donors worldwide at April 2013 indicate that there were 571,318 unrelated UCB units banked from 31 countries. To date, more than 10,000 patients have received an UCB transplant worldwide.

In parallel to these public banks, an entire industry has developed. These private (for-profit) banks improve, for a fee, collection of blood samples and cryopreservation of UCB units for a future use in the child or a related family member for related or allogeneic transplantation. While public banks use well established criteria for UCB units storage, private banks generally store all collected units, leading to a less quality parameter than in public banks [START_REF] Sun | Differences in quality between privately and publicly banked umbilical cord blood units: a pilot study of autologous cord blood infusion in children with acquired neurologic disorders[END_REF].

To meet the increasing demand in therapeutic us of UCB, the number of Public UCB banks is growing. There are now nearly 142 public banks versus 134 private banks actively involved around the world with more than 780,000 units [START_REF] Ballen | Collection and preservation of cord blood for personal use[END_REF].

The total number of UCB units in private banks exceeds largely the number conserved in the public ones. Conversely, the units from the former are the mostly used for transplantation. In addition, the probability for a child to be transplanted by his own cells ranges from 1:2,500 to 1:200,000 if these are not available for his family members [START_REF] Sullivan | Banking on cord blood stem cells[END_REF]. Indeed, more than 180 autologous UCBT has been successfully reported by private banks for non-malignant diseases (Cord Blood Registry in USA), the use of UCB units in malignant conditions being unlikely limited by the presence of pre-leukemic cells [START_REF] Greaves | Pre-natal origins of childhood leukemia[END_REF].

Also, the success rates of sibled UCB transplants from a family member are well defined than of transplants from a public donor's UCB and this is more important for ethnic populations [START_REF] Gluckman | Eurocord, Netcord, World Marrow Donor Association and National Donor Program. Family-directed umbilical cord blood banking[END_REF].

Many ethical debates have sparked regarding conflicting interests between private banks and the public ones and this public-private divide appears to have arisen because of the economic challenges of the UCB banks. Private banks encourage parents to store their child's UCB as a form of biological insurance for the child himself or a family members.

In these companies, communication and marketing efforts are made to open up this use to a large part of population.

Therefore, these banks are first created over the uncertainly future potential use of preserved own's child stem cells and the state of scientific knowledge [START_REF]) : Avis 74-Les banques de sang de cordon ombilical en vue d'une utilisation autologue ou en recherche[END_REF]. The likelihood that collected cells will be used in HSCT for established therapeutic indications is very low. Autologous UCB cells couldn't be used for genetic diseases because they carry the same genetic disease, and when required HSC are largely harvested from BM or PB [START_REF] Edozien | NHS maternity units should not encourage commercial banking of umbilical cord blood[END_REF][START_REF] Ballen | Collection and preservation of cord blood for personal use[END_REF].

Meanwhile, the current development of public sector policies regarding stem cells manipulation and with the increasing evidence for more therapeutic use of UCB stem cells outside of transplantation (as in regenerative medicine or in immune modulation) arise attractiveness to the private banks [START_REF] O'connor Ma 1 | Umbilical cord blood banking: beyond the public-private divide[END_REF][START_REF] Walbdy | Umbilical cord blood from social gift to venture capital[END_REF] and approaches to accommodate between public banks and personal interest in cord blood are unlikely to be achieved.

Even if private banks have developed extent procedures for collection and storage, and benefit rapidly from advanced biotechnology, their emergence create disagreements about the future lack of UCB units in public banks to provide treatments of some malignant diseases such as leukemia and lymphoma, and more especially about health care systems democratization and ownership of human tissues. Moral questions about how designing a health care system ensuring wellbeing, commitment of generational and social fairness, social solidarity with developing these emerging tissue economies should have answers [START_REF] Stewart | Ethics and law[END_REF].

Hybrid banks are gaining popularity in some countries, and companies are collecting UCB units for family personal use and at the same time other units for unrelated use (national). Many private banks now offer UCB to matched family members of a child and include them in research projects. So, the line between private and public banks is likely to be blurred. seem to be a functional model ensuring therapeutic and economic challenges [START_REF] O'connor Ma 1 | Umbilical cord blood banking: beyond the public-private divide[END_REF].

LEGISLATION AND BIOETHICS

Most experts agree that public UCB banking is the only option to obtain a suitable transplant to a vast number of people specifically for heterogeneous populations. These public banks are typically funded in part or entirely by public funds and their primary goal being the creation of an inventory of UCB units for unrelated use. They are envisaged either for solidarity and fair access to healthcare treatment purposes.

The USA and the European Union have not only regulated the therapeutic use of UCB for cell therapies and regenerative medicine, but also the banking of UCB units and the manufacture of manipulated stem cells. At the European level, cell therapy products are considered as medicinal products or as industrially produced medicinal products according to drug registration of innovative therapies system (MTI). They can also be considered as cell therapy preparations implemented in cell therapy units without the status of pharmaceutical establishment. Most of these countries have published statements on UCB and specific documents from the competent health authority, such as" l'Agence de biomédecine in France" or the FDA (Food and Drug Administration). Standards and specific international accreditations are drawn for all the process (from the collection to the distribution for transplantation). Since 2004, the main firms of biotechnology launched in applications on muscular dystrophy, bone damage, cartilage, type I diabetes, mesenchymal stem cells, diseases of the central nervous system and regulatory T cells.

There is a general consensus and a collaborative effort to establish international guidelines for better use of UCB and cell products for cell therapies. The NetCord-Fact international standard [START_REF]Standards for cellular therapy product services[END_REF][START_REF]International standards for cord blood collection, processing, testing, banking[END_REF] is now the reference for inspection and accreditation programs for standards cord blood collection, banking and release for administration.

The European Group on Ethics in Science and New Technologies has adopted a position on the ethical aspects of the private cord blood banking. Even autologous or family sibling banking is unlawful in some countries, some of them (Italy and Spain) regulate transport and storage in foreign accredited private banks in Switzerland and England where public banks coexist with the private ones.

In contrast to Europe where there is a strong bias in favor to public banks [START_REF] Katz | Banking cord blood stem cells : attitude and knowledge of pregnant women in five European Countries[END_REF][START_REF] Edozien | NHS maternity units should not encourage commercial banking of umbilical cord blood[END_REF], United States are less critical of private banking despite of the negative position of some professional organizations such as the American Academy of Pediatrics (AAP) [START_REF]Cord blood regulations, by state[END_REF]. In Australia, UCB transplantation is restricted to recognized applications and specific ethical issues are associated with collection, storage and access [START_REF] Petrini | Umbilical cord blood collection, storage and use : ethical issues[END_REF]. The Canadian Blood Services provide UCB units for banking and research in parallel to existing private banks [START_REF] Isasi | Fostering public cord blood banking and research in Canada[END_REF] In some countries of Asia, UCB stem cell banking is permissible and all UCB banks are regulated strictly as the blood banks such as in India. In China, there is only one public bank whereas in Japan and Korea private banks are dominants [START_REF] Petrini | Ethical issues in umbilical cord blood banking : a comparative analysis of documents from national and international institutions[END_REF]. In the absence of a clear regulation to govern the status of stem cell banks, some Eastern Arab countries banks are created as agents to well-known banks outside (as Future Health, Cells4Life, Cytocare,...). In Iran, public bank provides UCB banking since 2012 with more than 3000 units. In some other countries such as Oman and Lebanon, public and private banks cohabit and provide tissue banking [START_REF] Jamali | Cord blood banking activity in Iran national cord blood bank : A two years experrience[END_REF]. Saudia Arabia, United Arab Emirates (UAE), and Qatar have recently establish importants programs for promoting medical innovations and create public funding to support bank creation and processing [START_REF] Matsumoto | More than oil and uprisings: Current developments in cord blood banking in the Arab world: Abstract[END_REF]. Jordan, Saudia Arabia and Qatar have currently private banks and undergo general regulations for the first and religious decrees for the laters while Banks in UAE are under specific laws [START_REF] Matsumoto | More than oil and uprisings: Current developments in cord blood banking in the Arab world: Abstract[END_REF].

In north Africa, ministry of public health of Algeria gave permission to UCB bank in 2010 where prelevment and processing of UCB units were performed since 2000 [START_REF] Belhadj | Bilan d'activité de la banque de sang de cordon ombilical sur une période de 10 ans[END_REF]. Ethical committee remains the official authority regulating all the process since prelevment to transplantation. Egypt has two private banks licensed to keep cord blood cells, and three other banks are waiting for license, but the practice of stem cell therapy is limited by the Ministry of health and population to only two publics centers of excellence [START_REF] Sleem | Ethical and practice Issues in stem cell Research and Therapy "Egyptian Situation[END_REF]. The Tunisian UCB bank follows the guidelines by well established national laws dealing deontological practices and ethics and is established in the Centre National de Transfusion Sanguine [START_REF] Mellouli | A cord blood bank in Tunisia: usefulness and feasibility[END_REF] .

MOROCCO: The first CB bank

In Morocco, the first hematology department was created in Casablanca in 1980 and it is still the only public facility where adult patients can be treated for hematological diseases. Pediatric hematological disorders and cancer are treated in two other units. Some patients are treated in private clinics located in Casablanca and Rabat. The public hospitals still have limited resources and rely heavily on nongovernmental organizations to care of these patients [START_REF] Benchekroun S 1 | Present and future of hematology and stem cell transplantation in Morocco[END_REF]. The number of equipped bed ward set up remain insufficient while therapeutic needs of stem cell transplantations are well identified in children and adult patients and matched related family donors are not usually found. Unfortunately, only two allogeneic bone marrow transplantations have been realized till now. In addition, the cost of the transplantation procedure, in case of presence of matched donors, couldn't be supported by all population in the society.

But, the bigest obstacle to implementation of a UCB bank remains the revision of the law of organ donation and clinical trial authorization. While the world highlights the scientific and therapeutic advancement of embryonic and adult stem cells, efforts of Moroccan health system promote most often allopathic treatments. This, is due in general to the non commitment of Moroccan laws to the international advancements in stem cell therapy and to many other medical devises. While infrastructures are evolving with regard to national and international standards quality, and high technologic equipments are available, laws regulating clinical applications in new fields are still lagging behind.

Legislation

The law No. 16-98 relating to organ donation does not allow the altruistic donation, even free. However, the prelevment of organs from dead brain person is finally allowed for the future organ bank. It remains that the anonymous collection of UCB, considered as "surgical waste" and it's ex-vivo manipulation and unrelated administration are simply prohibited, even if the therapeutic need is justified (in sibled family in the case of hematological malignancies).

Ethic between the social benefit and the financial profit

The issues related to the creation of a stem cell bank, revealed by the feasibility study carried out by our laboratory since 2008 using a questionnaire distributed in the maternity of the Centre Hospitalier Universitaire (CHU) Ibn Rochd in Casablanca (In press) are multiples and importants.

Clinicians interested in these new therapies could more easily find stem cells or other cell products for some transplants in the absence of organ donors and an established organ bank (example type I diabetes). Areas of medical applications are multiples and concern oncology, endocrinology, kidney disease, severe burns, immunotherapy, toxicological studies, muscular dystrophy and neurodegenerative diseases. These applications can generate significant financial returns provided that adequate legislation is in place and a good financial model is identified.

The existing national ethic committee created by ministerial decree have just an advisory role and couldn't have an executive action especially with no specific drawn rules. However, it's main and first goal is to make cell therapies accessible for all the population.

RECOMMANDATIONS

Morocco should follow some guidelines and recommendations for its attractiveness and competitiveness in the field of UCB cell therapy. These recommendations focus on the legal and technical aspects of the use of stem cells or their derivatives in therapy and should be proposed in official texts.

To facilitate integration of cell therapy, several critical points should be considered:

-Allow altruistic donations of UCB cells and regulating their prelevment and transport to the processing laboratory;

-Allow expansion of ex-vivo stem cells and the production of manipulated stem cells for cell therapy;

-Create the institution involved in the manipulation and conservation of the stem cells derived UCB units and derivatives, and authorize their marketing through the creation of a national UCB bank; -Promote a public bank but allowing private transplant centers in the same way as the public one's to practice stem cell therapies;

-Allow biomedical research with cell therapy products derived from UCB and cord matrix by conducting trials for clinical validation of new treatment protocols and authorize the use of these products in public and private institutions approved in advance by the Ministry of Health; -Authorize the export of these UCB units to international transplant centers to be used as unrelated donor transplant to our residents abroad (MRE); -Adopt a policy of pricing and reimbursement of these UCB units and derivatives to encourage public institutions to carry out treatments and projects in clinical phase.

There is also evidence for the creation of a group of experts which can play a role of a regulatory agency rule on requests, for authorizations of advanced UCB stem cell therapy for biomedical research implementation. This group may also give

Table 1 : Umbilical Cord Blood Transplantation in Children with heamatopoietic malignancies Reference Number Age or median Degree of HLA match Neutrophil engraftment median or % Platelet engraftment median or % Acute GVHD I-II (%) Acute GVHD III-IV (%) Chronic GVHD Overall survival %

 1

	Rocha et al.	113	5 (<1-15)		26 days	44	29/107 patients 2/107 patients	5/93 surviving	46 (2-3 years)
	1990-1997 (19)			6/6
	Wagner et al.	13	2,7 (0,1-21,3) 6/6 , 1-2/6, 3/6 or	24 (16-53)	54 (39-130)	16	11	na	65 at 6 months
	1994-1995 (23)			more
	Dalle et al.			6/6, 1-2/6	28 (16-49)	43 (18-59)	3/36 (6 without	11	3/36 surviving	59 (2-3 years)
	1996-2002	30	7,5 (0,1-19,5)				hematopietic
	(3)						malignancies
	Rocha et al.	99	6 (2,5-10)	6/6 , 1-2/6, 3/6 or	32 (11-56)	81 (16-159)	38	21	5/43 patients	35
	1994-1998 (2)			more
	Kutzberg et al	191	7,7		27	174	Na	42	21% at 2 years	51 (1 year)
	2008 (25)
	Smith et al. 2009	21	4,2		95%	na	Na	1,2	0,8 (relative risk	43 (5 years)
	(26)								95% CI)
	Yoo KH et al.	236	na		18	45	Na	41,1	36,10%	47,5 (5 years)
	1996-2003 (27)							(Grade II-IV)
	Kato et al.	332	5		89% day 90	na	na	46	19%	38 (5 years)
	2011 (28)
	Yi E.S et al	41	9,1 (0,9-18,7)		19	51	Na	24	9%	na
	2005-2010 (21)							(Grade II-IV)
	Na : not available CI : confidence Interval

Table 2 : Umbilical Cord Blood Transplantation in Adults with heamatopoietic malignancies

 2

	Reference	Number	Age or	Degree of HLA	Neutrophil	Platelet	Acute GVHD	Acute GVHD	Chronic	Overall survival %
			median	match	engraftment	engraftment	I-II (%)	III-IV (%)	GVHD
					median or %	median or %
	Rocha et al.	98	24,5 (15-55)	6/6, 2/6, 3/6	26 (14-80)	na	13	13	18/16	63 (2-3 years)
	1998-2002								surviving
	(29)
	Wagner et al.	65	7,4 (0,2-56,9)	6/6, 2/6, 3/6	23 (9-54)	86 (29-276)	51/63 patients	17	9/59 patients	58 (1 year)
	1994-2001 (8)
	Langhlin et al.	150	16-60		27 (25-29)	60 (54-71)	In 150 patients		35/69	19 (2-3 years)
	1996-2001			2/6			Grades II, III,
	(30)						IV : 41
	Barker et al.	23	24 (13-53)	Double UCB trial :	23 (15-41)	na	52	13	5/23 patients	57 (1 year)
	2000-2003			4-6/6 matched each
	(4)			cord and the
				recipient
	Ballen et al	19	49 (24-63)	Double UCB trial :	20 (15-34)	41 (21-55)	53	5	5/16 patients	71 (1 year)
	2003-2005			4-6/6 matched to
	(15)			each other and to
				the recipient
	Brunstein et	106	51 (17-69)	Double UCB trial :	12 (0-32)	49 (0-134)	37/110 patients	22/110	23/110	45 (3 years)
	al.			4-6/6 matched to			(4 without
	2001-2005			each other and to			hemato
	(16)			the recipient			malignancies)
	Takahashi et	100	38 median		85%	na	Na	52 (Grade II-	71%	70 (3 years)
	al. 2007 (12)							IV)
	Kumar et al.	173	38 median		Na	na	Na	32 (Grade II-	28%	36 (2 years)
	2008 (31)							IV)
	Astuta et al.	114	34 median		Na	na	Na	28 (Grade II-	27%	45 (2 years)
	2009 (13)							IV)
	Brunstein et	351	37 median		0,5(0,42-0,6)	na		0,55 (0,42-	1,36% (0,99-	0,97 (0,92-1,35)
	al. 2010 (17)							0,72)(Grade	1,88)
								II-IV)
	Na : not available

Table 3 : Umbilical Cord Blood Transplantation in Patients with non heamatopoietic malignancies

 3

	Reference	Disease	Number	Age or	Neutrophil engraftment	Chronic GVHD	Overall survival
				median	median or %
							%
		Inherited metabolic disorders	159	na	Na	na
	Prasad et al.	-Hurler Syndrome	45				77 (1 year)
	2008 (32)	-Metachromatic Leukodystrophy	15				65 (1 year)
		-Krabbe Diseases	36				74 (1 year)
		-Sanfilipo Syndrome	19				79 (1 year)
		-Adrenoleukodystrophy	13				77 (1 year)
	Frangaul H. et	Primary immune Deficiencies	364	0,2-7,8	58% (day 42)	na	63 (1 year)
	al
	1999-2003
	(33)
	Bizzetto et al	Hereditary BM failure	64	na	95% of R CBT	Acute GVHD II-IV : 2/20	95 (3 years)
	2011					Chronic 11% (2 years)
	(34)
						Acute GVHD II-IV : 24%	61 (3 years)
					86% of UR CBT	(day 100)
						Chronic GVHD 53% (2 years)
	Na : not available
	R CBT : Related Cord Blood Transplantation
	UR : Unrelated Cord Blood Transplantation

STATE OF THE ART OF UCB BANKING: Public or Private?

Journal of Stem Cells & Regenerative MedicineVol 10 No2, 2014 : 28-37

an opinion on the changes in biomedical research using stem cell products, or in case of serious side effects observed after administration of the therapeutic. Steering committees can also be created with various objectives and timelines. This committee will define and approve new clinical investigations related to national priorities in terms of public health, cost and ethics.

CONCLUSION

Our principal goal is to consider the therapeutic benefit of the stem cell products and derivatives and to make stem cell therapy accessible to Moroccan people. On the other hand, many misconceptions about donation of organs or blood are conducted by illiterate and ignorant people and getting doubt about what is going to be done with them. In Arabic and Muslim cultures, people are educated to make offerings of their property, money and time, but never from himself, even if the religion does not prohibit this type of donation. So, considerable efforts have to be done on educating large numbers of people and especially parents regarding UCB use and laws by developing communications and networking between authorities, professional associations, physicians, non-governmental organizations and opinion leaders.

DISCLOSURE

The second and third author have participated equally to this work.


Pierre-Arnaud Coquelin
email: coquelin@cmapx.polytecnique.fr

Sophie Martin
email: sophie.martin@clermont.cemagref.fr

Rémi Munos
email: remi.munos@inria.fr

A Dynamic Programming Approach to Viability Problems

Viability theory considers the problem of maintaining a system under a set of viability constraints. The main tool for solving viability problems lies in the construction of the viability kernel, defined as the set of initial states from which there exists a trajectory that remains in the set of constraints indefinitely. The theory is very elegant and appears naturally in many applications. Unfortunately, the current numerical approaches suffer from low computational efficiency, which limits the potential range of applications of this domain.

In this paper we show that the viability kernel is the zero-level set of a related dynamic programming problem, which opens promising research directions for numerical approximation of the viability kernel using tools from approximate dynamic programming. We illustrate the approach using k-nearest neighbors on a toy problem in two dimensions and on a complex dynamical model for anaerobic digestion process in four dimensions.

I. INTRODUCTION

In a typical viability problem, the state of a controlled dynamical system is submitted to viability constraints. Solving the problem means discriminating the states from which there exists at least one control law such that the system remains in the constraint set indefinitely (these state define the viability kernel), from the states from which the system is doomed to go beyond the viability constraints in finite time.

To solve such problems, viability theory has been designed and developed for about twenty years [START_REF] Aubin | Viability theory[END_REF]. It encompasses a great deal of theoretical work and provides efficient mathematical tools for analyzing the local and asymptotic behavior of control systems. It is also used to study the evolutions of uncertain systems subjected to viability constraints arising in socio-economic and biological sciences, as well as in control theory, since their expression is often natural under this formalism (see for instance [START_REF] Béné | A viability analysis for a bioeconomic model[END_REF] for the management of a renewable resource, [START_REF] Mullon | Viability model of trophic interctions in marine ecosystems[END_REF] for the analysis of fishing quotas consequences, and [START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF] for a lake eutrophication problem in ecology).

Formally, a typical viability problem is defined as follows. Consider a control system, defined by the differential equation: ẋ(t) = f (x(t), u(t)), u(t) ∈ U for almost every t ≥ 0 x(0) = x 0 [START_REF] Aubin | Viability theory[END_REF] where f : R d × R p → R d is the state dynamics and U a compact subset of R p , the control space. We denote by F the set-valued map defined by F (x) := {f (x, u), u ∈ U }, for every x ∈ R d .

A viability problem considers evolutions x(•) that are viable in a subset K ⊂ R d representing an environment in which the trajectory of the evolution must remain forever: ∀t ≥ 0, x(t) ∈ K.

The Viability Kernel of the subset K under the dynamics F , denoted by Viab F (K), is defined as the set of points x 0 from which can start a viable solution, i.e., Viab F (K) := {x 0 ∈ K, ∃u(•) ∈ L ∞ ((0, +∞); U ), such that the solution of (1) associated to u(•) satisfies: ∀t ≥ 0, x(t) ∈ K} .

The viability kernel may be characterized in diverse ways through tangential conditions thanks to the viability theorems [START_REF] Aubin | Viability theory[END_REF]. The computation of the viability kernel also serves for designing interesting feed-back policies, such as the heavy controller which remains constant as long as possible until the boundary of the viability kernel is reached. There are several numerical approaches for approximating the viability kernel. The viability kernel algorithm, proposed by Saint-Pierre [START_REF] Saint-Pierre | Approximation of viability kernel[END_REF], computes, for a given discretization step ∆h of the space and time, a grid-based representation of the viability kernel Viab ∆h F (K) such that: lim ∆h→0 Viab ∆h F (K) = Viab F (K). Bokanowski et al. [START_REF] Bokanowski | An anti-diffusive scheme for viability problems[END_REF] characterized the viability kernel by the infinite time limit of the value function of an evolutionary control problem and this value function being discontinuous, used the anti-diffusive Ultra-Bee numerical scheme extended to the resolution of Hamilton-Jacobi-Bellman equations to reduce numerical diffusion.

These approaches share common features that limit their applicability to real world problems: the discretization in space is grid-based and therefore subject to the curse of dimensionality: the computational time and memory needed to find and store Viab (K) such that d(Viab (K), Viab(K)) ≤ grow exponentially with the dimension of underlying state space. Consequently these algorithms do not work in dimensions higher than 3. More efficient subset representation is considered by Chapel et al. [START_REF] Chapel | Defining yield policies in a viability approach[END_REF] where Support Vector Machines summarize the information contained in the grid and provide an analytical description of the viability kernel.

In this paper we address the issue of efficiently computing the viability kernel using approximate dynamic programming (see eg. [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF]) methods. We propose an original approach that proceeds in three steps. First, we characterize Viab F (K) as the zeros of a [0, 1]-valued map V * (x) geometrically decreasing with the maximal exit time of K when starting at x ∈ K. Second, we show that V * solves the Hamilton-Jacobi-Bellman equation, that we discretize in time using a finite time-step ∆t, and use an approximate dynamic programming algorithm to approximate the solution V ∆t of the resulting Bellman equation. Third, we define the approximate viability kernel associated to the function V ∆t by the under-level set K := {x ∈ K, V ∆t (x) ≤ } with > 0 a given threshold.

A contribution of this work is to establish a link between the viability kernel and the under-level set of an approximate solution of a related dynamic programming (DP) problem. Our purpose in this paper is to describe how one may use approximate dynamic programming for numerically solving viability problems. In particular, we believe this would enable to introduce machine learning tools such as statistical learning (see e.g. [START_REF] Hastie | The Elements of Statistical Learning[END_REF]) and function approximation (see e.g. [START_REF] Devore | Nonlinear Approximation[END_REF]) into the viability community. Another contribution is the viability study of the anaerobic digestion process model described in [START_REF]IWA Task Group for Mathematical Modelling of Anaerobic Digestion Processe[END_REF]. We illustrate the fact that the computation of the viability kernel provides a very natural yet powerful tool to analyze, understand, and control the complex behavior of this chemical process.

The paper is organized as follows. Section II establishes the connection between the viability kernel and the zerolevel set of the value function for a related DP problem. We also mention a convergence result for the time-discretized problem. In Section III, we describe the numerical method for solving the discrete-time DP problem and illustrate the method using Approximate Value Iteration combined with k-nearest neighbors. We recall an error bound for this ADP method in terms of L ∞ norm. In Section IV we bound the approximation error of the viability kernel in terms of the approximation power of the function space. In Section V we present the numerical examples in two and four dimensions that illustrate the approach.

II. A DYNAMIC PROGRAMMING APPROACH TO SOLVING VIABILITY PROBLEMS

In this section we establish the link between the viability kernel and the zero level set of the value function of a related Dynamic Programming (DP) problem. We then introduce a consistent time-discretization procedure whose solution will be numerically approximated in the next section.

First, let us define τ : R n × L ∞ (R + ; U ) → R + as the exit time from K of the trajectory x(•) (defined by (1)), starting from x 0 under the control law u(•): τ

(x 0 ; u(•)) := inf{t ∈ R + |x(t) /
∈ K} with the convention that τ = ∞ if the trajectory stays forever in K. Then, the maximum value τ * : R n → R + of the exit time, for all possible controls, is

τ * (x 0 ) = sup u(•)∈L ∞ (R+;U ) τ (x 0 ; u(•)) .
From its definition, the viability kernel is the set of points with infinite exit-time value:

Viab F (K) = {x ∈ K, τ * (x) = +∞}.
We now introduce the related discounted optimal exit time control problem. The discount factor is denoted by γ ∈ (0, 1) and the optimal value function V * : R n → R is defined by:

V * (x) := inf u(•)∈L ∞ (R+;U ) [γ τ (x,u(•)) ] = γ τ * (x) ,
which is 1 discounted by γ to the power the maximum exit time.

It is straightforward to check that the viability kernel is the zero level set of

V * : Viab F (K) = {x ∈ K, V * (x) = 0}.
Consequently, computing the value function V * enables to deduce the viability kernel. This is the basic idea of the dynamic programming approach to compute viability kernels.

The optimal value function solves a non-linear partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation in the sense of viscosity solutions (see eg. [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF]): if

V * is differentiable at x ∈ K, then V * (x) log(γ) + min u∈U ∇V * (x) • f (x, u) = 0, (3)
with the (boundary) condition V * (x) = 1 for x / ∈ K. The analysis of such equations and the formalism of viscosity solutions is beyond the scope of this paper; we refer the interested reader to the references [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF], [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]. Now, we discretize in time the HJB-equation using some consistent and monotonic numerical scheme. Let ∆t be the time-step, and write V ∆t the approximate value function, which is the unique solution of the following dynamic programming equation: for x ∈ K,

V ∆t (x) = γ ∆t min u∈U V ∆t (g ∆t (x, u)) (4)
where g ∆t (x, u) := x + f (x, u)∆t is the (Euler-based) discretized state-dynamics. The boundary condition is the same: V ∆t (x) = 1 for x / ∈ K. The consistency of this numerical scheme holds since a regular solution V of (3) plugged in the scheme (4) would yield a truncation error of order 1 in time, i.e.

min u∈U γ ∆t V (x + f (x, u)∆t) -V (x) ∆t = min u∈U ∇V (x) • f (x, u) + V (x) log(γ) + O(∆t) = O(∆t)
Now, in order to guarantee the convergence of the discretetime value function V ∆t to the optimal value function V * , when ∆t tends to zero, we make use of the general results of [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF], where the comparison principle (between lower and upper viscosity solutions) is a consequence of the following assumption on the state dynamics at the boundary:

Assumption 1: Note -→ n (x) the outward normal of K at x ∈ ∂K. We assume that for all x ∈ ∂K,

if ∃u ∈ U , s.t. f (x, u) • -→ n (x) ≤ 0, then ∃u ∈ U , s.t. f (x, u) • -→ n (x) < 0, if ∃u ∈ U , s.t. f (x, u) • -→ n (x) ≥ 0, then ∃u ∈ U , s.t. f (x, u) • -→ n (x) > 0.
Under this assumption, we have the convergence result (see [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] for a proof):

lim ∆t→0 V ∆t = V * .
It is worth noting that the technical assumption 1 guarantees that the optimal value function V * is continuous, which simplifies the convergence proof of numerical schemes such as (4). However, in general this assumption is by no means necessary for convergence, but without it, the proof become more tedious and requires the theory of non-continuous viscosity solutions (see eg. [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]).

Let us introduce the viability kernel Viab ∆t g (K) of K under the discrete dynamics g ∆t . It is also straightforward to check that Viab ∆t g (K) is the zero-level set of V ∆t that is to say: Viab ∆t g (K) = (V ∆t ) -1 (0). Thus in the sequel, we aim at computing an approximation of the discrete-time solution V ∆t for a given (small) ∆t.

For notation clarity, we omit in the remainder of the paper the ∆t exponent, simply writing V for V ∆t , g for g ∆t , Viab g (K) for Viab ∆t g (K) and γ for γ ∆t . III. NUMERICAL METHODS AND ERROR BOUNDS FOR THE DISCRETE-TIME DP PROBLEM We may write the DP equation ( 4) as a fixed-point equation V = T V where T is the DP operator, defined, for any bounded function W over K and for all x ∈ K, by :

T W (x) := γ min u∈U W (g(x, u)) if g(x, u) ∈ K 1 if g(x, u) / ∈ K
Since T is a contraction mapping in • ∞ norm, it admits a unique fixed-point that can be computed by regular DP methods (see eg. [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF]) such as the value iteration algorithm:

• Initialization: start with any V 0 ∈ R K • Iteration: repeat V n+1 = T V n until convergence occurs (see above references) Unfortunately, in continuous state spaces or in large discrete spaces, the value function cannot be computed exactly, and has to be represented using a finite number of parameters. In this paper we investigate function approximators where the values are represented using a finite set of states

X N := {x i } 1≤i≤N ⊂ K.
Let us define an approximation operator A N : R K → R K , based on X N , such that for any function

W ∈ R K , A N (W ) is defined only from the N values {W (x i )} 1≤i≤N . We write F N the image of R K by A N , ie: F N := A N (R K ).
Examples of such operators include k-nearest neighbors, locally weighted regression, linear approximation, and kernel methods in reproducing kernel Hilbert spaces, such as SVMs (see eg. [START_REF] Hastie | The Elements of Statistical Learning[END_REF]). The one-nearest neighbor approximator, defined by A N (W )(x) := W (arg min y∈X N x -y ) satisfies the uniform approximation bound

W -A N (W ) ∞ ≤ (F N ) (5)
in the class of L-Lipschitz functions on K (ie. functions W such that ∀x, y ∈ K, |W (x) -W (y)| ≤ L x -y ). Indeed, for such functions, we can take (F N ) = Ld(X N , K) where

d(X N , K) := sup x∈K min 1≤i≤N x -x i is the density of X N in K.
The Approximate Value Iteration (AVI) algorithm is defined by (see e.g. [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF]):

• Initialization: Sample N states X N = {x i } 1≤i≤N and choose any function V 0 ∈ R K . • Iteration: repeat V n+1 := A N (T V n ) until some stopping criterion is met (see [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF]). Now, assuming that the approximation bound (5) holds for functions in T F N (thus, at each iteration, the approximation error satisfies V n+1 -T V n ∞ ≤ (F N )), then a bound on the approximation error of the value function V , in terms of the approximation capacity of F N is:

V -V n ∞ ≤ 1 1 -γ (F N ) + γ n V -V 0 ∞ , (6)
(which simply results from an inductive argument based on the inequality

V -V n ∞ ≤ T V -T V n-1 ∞ + T V n-1 - A N (T V n-1 ) ∞ ≤ γ V -V n-1 ∞ + (F N )).

IV. APPROXIMATION OF THE VIABILITY KERNEL

In this section we explain how to derive an approximate viability kernel by selecting the states whose approximate value function is below a given positive threshold. This thresholding procedure is reminiscent of the two-classes classification problem where a potential function h is trained based on labeled data, and then used for the prediction at a new state x based on sign(h(x) -b) (where b is the threshold). Viability kernel approximation using sequences of SVM classification problems has been undertaken in [START_REF] Chapel | Defining yield policies in a viability approach[END_REF].

In some way, this paper illustrates that a good choice of potential function h for viability kernel classification is the value function of the related dynamic programming problem introduced in the previous section.

For any function h and real number η > 0, let us defined the under-level set K η (h) = {x ∈ K, h(x) ≤ η}. We have the following result concerning the approximation of the viability kernel K by K η ( V ), where V is an approximation of the value function.

Theorem 1: Assume that there exists two constants C > 0 and α > 0 such that for all x in a small neighborhood of Viab g (K), V (x) ≥ Cd(x, Viab g (K)) α . Assume that the approximate value function V is η-close to V , i.e. V -V ∞ ≤ η. Then the approximate viability kernel K η ( V ) satisfies, for small η > 0,

Viab g (K) ⊂ K η ( V ) ⊂ K 2η (V ) ⊂ Viab g (K)+B 0, (2η/C) 1/α .
where B(0, δ) is the (euclidian) ball centered at 0 with radius δ.

The assumption in Theorem 1 expresses a lower bound on the increase of V in the direction of the outward normal of the viability kernel, and is not very stringent in general. A typical value α = 2 is usual, although one may build artificial problems for which this increase is exponentially slow (all derivatives being equal to zero).

Proof: This result derives from the definition of the underlevel sets. In short, if x ∈ Viab g (K), then

V (x) = 0 thus V (x) ≤ η thus x ∈ K η ( V ). Now if x ∈ K η ( V ), then V (x) ≤ V (x) + η ≤ 2η, thus x ∈ K 2η (V ). Finally, if x ∈ K 2η (V ) then d(x, Viab g (K)) ≤ ( 2η
C ) 1/α thus (for small η) x ∈ B 0, ( 2η C ) 1/α . As a consequence, one may derive a convergence result to the viability kernel of the continuous time problem Viab F (K) (defined by ( 2)).

Corollary Let V N be a sequence of approximate value functions computed by the method of Section III, using a decreasing time-step ∆t(N ), such that the approximation error satisfies (F N )/∆t(N ) → 0 when N → ∞ (which is the case for k-nearest neighbors with uniformly sampled states X N with a ∆t(N ) decreasing strictly more slowly than N -1/d , where d is the space dimension), then

lim N →∞ K 1/N (V N ) = Viab F (K).
This consistency result is a direct consequence of Theorem 1 and guarantees that for a rich enough class of function approximators, the approximate viability kernel defined by the under-level set of the resulting approximate value function is close to the true viability kernel.

Proof: From the hypothesis, for any N , there exists (F N ) and ∆t(N ) such that

(F N )/∆t(N ) ≤ 1/N . Now, from (6), V * -V N ∞ ≤ 2 (F N )/(1 -γ ∆t(N ) ) for large enough N , thus V * -V N ∞ ≤ 4 (F N )/∆t(N ) ≤ 4/N ,
and the convergence follows as a direct consequence of Theorem 1. In the case of k-nearest neighbors with uniformly sampled states, the density of points ( (F N ) as mentioned in the previous section) is of order 1/N 1/d . Thus if ∆t(N ) decreases strictly more slowly than N -1/d , then (F N )/∆t(N ) → 0 for N → ∞.

V. SIMULATIONS

A. A two-dimensional consumption problem

We illustrate the method on this two-dimensional consumption problem (see [START_REF] Aubin | Viability theory[END_REF], [START_REF] Saint-Pierre | Approximation of viability kernel[END_REF]), defined by the state dynamics

ẋ(t) = x(t) -y(t) ẏ(t) = u (7)
with the control u ∈ U := {-1 2 , 1 2 }, and the constraints x(t) ∈ [0, 2] and y(t) ∈ [0, 3]. Viability kernels make it possible to define interesting control policies. Let us simply illustrate the notion of timorous controller, which is a secure variant of the so-called heavy controller (see eg. [START_REF] Aubin | Viability theory[END_REF]) where the control is kept constant until the state is below a fixed (security) distance to the boundary of the viability kernel. When this happens, the chosen control is the one maximizing the distance from the next resulting states to the boundary. This timorous controller provides a robust, secure, and piecewise constant controller. A piece of trajectory following a timorous controller (with a security distance of 0.05) is shown in black line in Figure 1 (right).

B. A four dimensional anaerobic digestion model

Anaerobic wastewater treatment process is a powerful and ecological process to degrade organic substrate. However, its instability prevents it to be used on a wide scale industrially. Consequently, during the last decades there has been a growing interest in modeling and studying its dynamical behavior [START_REF]IWA Task Group for Mathematical Modelling of Anaerobic Digestion Processe[END_REF], [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF], aiming at building stable controllers [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF].

Anaerobic digestion can be viewed as a two stage chemical process:

Acidogenesis: k 1 S 1 µ1(S1)X1 -→ X 1 + k 2 S 2 + k 4 CO 2 Methanogenesis: k 3 S 2 µ2(S2)X2 -→ k 5 CO 2 + k 6 CH 4
where k 1 , . . . , k 6 are the stoechiometric constants and S 1 , S 2 , X 1 , X 2 are respectively the concentration of organic substrate, of volatile fatty acid, of acidogenesis bacteria, of methanogenesis bacteria in the tank. The reaction rates µ 1 (•) and µ 2 (•) are given by:

µ 1 (S 1 ) = µ 1max S 1 S 1 + K S1 µ 2 (S 2 ) = µ 2max S 2 S 2 + K S2 + S 2 2 /K I2 .
where µ 1max , µ 2max , K S1 , K S2 , and K I2 are constants. By a mass balance calculation, one may derive the following 4dimensional dynamical model:

Ẋ1 = (µ 1 (S 1 ) -αD)X 1 , Ṡ1 = D(S 1in -S 1 ) -k 1 µ 1 (S 1 )X 1 , Ẋ2 = (µ 2 (S 2 ) -αD)X 2 , Ṡ2 = D(S 2in -S 2 ) -k 2 µ 1 (S 1 )X 1 -k 3 µ 2 (S 2 )X 2 ,
where α is the rate of bacteria in suspension in the liquid phase, S 1in , S 2in are the effluent characteristics. D ∈ {0, 0.5} is the liquid flow per volume unit and represents the control parameter (which takes two possible values). Since those dynamics modelize an unstable phenomenon, the knowledge of viability states is of particular interest. The main cause of instability is the accumulation of volatile fatty acid S 2 in the tank which inhibits the methanogenesis phase, and then produces even more S 2 which eventually acidifies the tank and kills all the bacteriological species. Thus, the natural constraint set is given by all variables being non-negative and S 2 is additionally required to be upper bounded, thus:

(X 1 , X 2 , S 1 , S 2 ) ∈ (R + ) 3 × [0, 10].
We compute the four dimensional viability kernel where we chose the same numerical values as in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. For the algorithm parameters, we took ∆ t = 0.005, γ ∆t = 0.999. We run n = 500 value iterations using a 3-nearest neighbor approximator using 2.10 5 points uniformly sampled in the domain K

= {(X 1 , X 2 , S 1 , S 2 ) ∈ [0, 2] × [0, 0.2] × [0, 3] × [0, 10]}.
The full computation of the viability kernel takes 162.52 sec. with Matlab on a 1.8 Ghz computer. Projections of the viability kernel onto two dimensional spaces are represented in Figures 2,3, and4. For clarity, we plotted the set of points located on a regular grid whose nearest neighbor belongs to the underlevel set of the approximate value function (with a threshold of 0.001).

We believe that the results are very interesting and provide an original visual representation of the stability problem for an anaerobic digestion. We observe three kinds of dependencies between variables directly from the geometry of the viability kernel boundary: convex dependency (Figure 2), linear dependency (Figure 3), and independency (Figure 4). The results are in accordance with the qualitative interpretation of the chemical phenomenon. For example, Figure 2 illustrates the fact that we cannot have at the same time a large amount of organic substrate S 1 and acidogenesis bacteria X 1 . Indeed this would induce a fast production of volatile fatty acid S 2 which would lead to the reaction death. The other figures also illustrate in a simple and intuitive way similar relationships between variables.

Moreover, the viability kernel contains quantitative relevant knowledge that can be exploited, for example, to design a timorous controller (as defined in previous section) which is particularly adapted here since it provides a safe controller with infrequent switches.

VI. CONCLUSION

This paper describes an alternative approach for solving viability problems by introducing a related exit-time discounted optimal control problem. We showed how the viability kernel is approximated by the under-level set of an approximate value function for some small positive threshold, and provide approximation error based on the capacity of the considered function space. To illustrate, we used a simple k-nearest neighbors approximator with states sampled uniformly randomly inside the set of constraints K.

Of course, more sophisticated function approximators, and other approximate DP algorithms (such as policy iteration based) could be used to improve performance. Besides, adaptive methods could easily be designed from this work. Indeed, once a rough approximate kernel K η (V N ) has been obtained, one may build a new problem by redefining the constraint set to be K η (V N ) instead of K. As long as Viab g (K) ⊂ K η (V N ) (which is the case if η ≥ (F N )), then we still have Viab g (K) = Viab g (K η (V N )). The benefit being that since K η (V N ) is smaller than K, sampling N states in this subset would yield a higher density of points and thus would decrease (F N ).

Another way of decreasing (F N ) is to remove the states that are strictly inside the current approximation of the viability kernel (e.g. that lie at a lower bounded security distance from the boundary). The main idea being that we only need to sample points around the boundary of the viability kernel.

Extension of this approach to other viability problems, such as approximating the capture basin is straightforward.

Figure 1

 1 represents (left) the approximate value function V n (here n = 200) obtained for a set X N of N = 10 5 points sampled uniformly over the domain K = [0, 2] × [0, 3]. The discount factor γ ∆t = 0.95 (with ∆t = 0.05). The approximator used here is the 3-nearest neighbors. The right figure shows the set of states {x i , V n (x i ) ≤ 0.01} being an approximation of the viability kernel.

Fig. 1 .

 1 Fig. 1. The consumption problem. Top: value function of the related DP problem. The variables x and y lie in the (x, y)-plane, and the the value of the value function for this problem is indicted in the z-axis. Bottom: set of points {x i } in the approximate kernel (such that Vn(x i ) ≤ 0.01) and a timorous trajectory starting from x 0 = 0.4, y 0 = 0.3 and using a security distance of 0.05.)

Fig. 2 .

 2 Fig. 2. Representation of S 1 versus X 1 for different values of S 2 . Notice the convex dependency between the variables.

Fig. 3 .

 3 Fig. 3. Representation of S 2 versus S 1 for different values of X 1 . Notice the linear dependency between the variables.

Fig. 4 .

 4 Fig. 4. Representation of S 1 versus X 2 for different values of S 2 . Notice the independency of S 1 w.r.t. X 2 .