N Auffray
email: nicolas.auffray@univ-mlv.fr

Analytical expressions for odd-order anisotropic tensor dimension

Keywords: Anisotropic materials, Tensors, Generalized elasticity Matériaux anisotropes, Tenseurs, Elasticité généralisée

come

Introduction

In the multiphysical modeling of anisotropic behaviors [START_REF] Thionnet | A new constructive method using the theory of invariants to obtain material behavior laws[END_REF] it is interesting to have prior information on the general characteristics of the sought model. The knowledge, for a given material anisotropy, of the number of independent coefficients in the tensorial law is interesting since, for example, it determines the number of elementary tests one needs to perform in a micro-macro homogenization procedure [START_REF] Trinh | Evaluation of generalized continuum substitution models for heterogeneous materials[END_REF]. These results are also important to develop strategies for the experimental identification of constitutive laws [START_REF] Francois | Une nouvelle analyse des symétries d'un matéeriau élastique anisotrope. Exemple d'utilisation à partir de mesures ultrasonores[END_REF][START_REF] Francois | Determination of the symmetries of an experimentally determined stiffness tensor: application to acoustic measurements[END_REF]. If in the case of classical elasticity these results are well-known [START_REF] Cowin | Eigentensors of linear anisotropic elastic materials[END_REF][START_REF] Forte | Symmetry classes for elasticity tensors[END_REF], their extension to other situations is not straightforward. However, due to a growing interest in the modeling of size and non-local effects in materials and structures, the need for generalized continuum theories becomes evident [START_REF] Fleck | Strain gradient plasticity[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF][START_REF] Sciarra | Second gradient poromechanics[END_REF]. This note, and following a path already developed in some previous contributions [START_REF] Auffray | Démonstration du théoréme d'Hermann é partir de la méthode Forte-Vianello[END_REF][START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF][START_REF] Auffray | Analytical expressions for anisotropic tensor dimension[END_REF], aims at providing some tools that can help the modeling of these non-conventional behaviors.

In the present paper, and following a previous one [START_REF] Auffray | Analytical expressions for anisotropic tensor dimension[END_REF] in which only even-order tensors were considered, we provide analytical formulas that give the generic dimension of any anisotropic odd-order tensor1 . To avoid any misunderstanding, it is worth noting that the present method does not solve the symmetry classification for a physical tensor, but only provide a way to compute, once the classification has been done, the number of generic coefficients for each anisotropic system. The solution of the classification problem can be found in the following references [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF]. As an illustration, the obtained formula are applied to the space of third-order piezoelectric tensor [START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Weller | Piezomagnetic tensors symmetries: an unifying tentative approach[END_REF], and to the fifth-order coupling tensor of Mindlin's strain-gradient elasticity [START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF].

This note is organized as follows, in Section 2 basic definitions about symmetries are summed up (see, e.g. [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF] for an extended discussion). The decompositions of a tensor space into orthogonal elementary components and the notion of G-invariant space are developed in Section 3, and our main formula are provided there. Finally, physical illustrations are proposed in Section 4.

Physical and material symmetries

Hereafter E 3 will be the 3D euclidean physical space. Let G be a closed subgroup of O(3) the orthogonal group in 3D, that is the group of isometries of E 3 . Let us define a material M as an open subset of E 3 . The set of operations Q ∈ O(3) leaving M invariant is defined as

G M = {Q ∈ O(3), Q M = M}
where stands for the Q action upon M. This set, denoted G M , is known as the material symmetry group. Now consider a physical property P defined on M, the set of operations leaving P invariant is the physical symmetry group G P = {Q ∈ O(3), Q P = P} P is described, in the present paper, by a n th -order tensor T (n) ∈ T (n) . In that case the action of O(3) on T (n) is defined by the Rayleigh product:

: O(3) × T (n) → T (n) : (Q, T (n) ) → Q T (n) := Q i1j1 . . . Q injn T (n) j1...jn (1)
The material and the physical symmetry groups are related by the mean of the Curie-Neumann's principle [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF]: G M ⊆ G P meaning that each operation leaving the material invariant leaves the physical property invariant. Nevertheless, as shown for tensorial properties using Hermann's theorems [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF], physical properties can be more symmetrical than the material.

In E 3 G P is conjugate to an O(3)-closed subgroup [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. Classification of O(3)-closed subgroups is a classical result that can be found in many references, e.g. [START_REF] Sternberg | Group theory and physics[END_REF]: Lemma 2.1 Every closed subgroup of O(3) is conjugate to precisely one group of the following list, which has been divided into three classes:

(i) Closed subgroups of SO(3): 1, Z n , D n , T , O, I, SO(2), O(2), SO(3); (ii) K := K ⊕ Z c
2 , where K is a closed subgroup of SO(3) and Z c 2 = {1, -1}; (iii) Closed subgroups not containing -1 and not contained in SO(3):

Z - 2n (n ≥ 1), D v n (n ≥ 2), D h 2n (n ≥ 2), O -or O(2) -
Let us now give a brief description of these different subgroups:

Type I subgroups Among SO(3)-closed subgroups we can distinguish: Let us detail first the set of planar subgroups. We fix a base (i; j; k) of R 3 , and denote by

Planar groups : {1, Z n , D n , SO(2), O (2)
Q(v; θ) ∈ SO(3) the rotation about v ∈ R 3 , with angle θ ∈ [0; 2π) we have • 1, the identity; • Z n (n ≥ 2)
, the cyclic group of order n, generated by the n-fold rotation Q(k; θ = 2π n ), which is the symmetry group of a chiral polygon;

• D n (n ≥ 2), the dihedral group of order 2n generated by Z n and Q(i; π), which is the symmetry group of a regular polygon;

• SO(2), the subgroup of rotations Q(k; θ) with θ ∈ [0; 2π). It is the symmetry group of an oriented cone.

• O(2), the subgroup generated by SO(2) and Q(i; π). It is the symmetry group of an twisted cylinder.

The classes of exceptional subgroups are: T the tetrahedral rotation group of order 12 which fixes a tetrahedron, O the octahedral rotation group of order 24 which fixes an octahedron (or a cube), and I the rotation group of order 60 which fixes an icosahedron (or a dodecahedron).

Type II subgroups

Type II subgroups are of the form K := K ⊕ Z c 2 , where K is a closed subgroup of SO(3). Therefore we directly know the collection of type II subgroups.

Type III subgroups

The construction of type III subgroups is more involved, and a description of their structure is provided in [START_REF] Sternberg | Group theory and physics[END_REF]. As for type I subgroups, we can introduce subgroups of type III. Let σ u ∈ O(3) denotes the reflection through the plane normal to u axis.

• Z - 2 the order 2 reflection group generated by σ i ;

• Z - 2n (n ≥ 2) the group of order 2n, generated by the 2n-fold rotoreflection

Q(k; θ = π n ) • σ k ; • D h 2n (n ≥ 2)
the prismatic group of order 4n generated by Z - 2n and Q(i, π). When n is odd it is the symmetry group of a regular prism, and when n is even it is the symmetry group of a regular antiprism;

• D v n (n ≥ 2)
the pyramidal group of order 2n generated by Z n and σ i , which is the symmetry group of a regular pyramid;

• O(2) -is the limit group of D v n for continuous relation, it is therefore generated by Q(k; θ) and σ i . It is the symmetry group of a cone;

• O -which is achiral tetrahedral symmetry of order 24. This group has the same rotation axes as T , but with six mirror planes, each through two 3-fold axes.

In order to have a better physical understanding of these subgroups, we reported in Appendix A the tables making correspondences between group notations and the classical crystallographic ones (Hermann-Mauguin, Schoenflies).

To study the symmetry classes of a tensor we need to decompose it into O(3)-elementary parts.

Structure of tensor spaces

Harmonic decomposition

The O(3)-invariant decomposition of a tensor is known as harmonic decomposition, it is an higherdimensional analogue of the Fourier decomposition. It allows to decompose any finite-order tensor into a sum of irreducible ones [START_REF] Zou | Orthogonal Irreducible Decompositions of Tensors of High Orders[END_REF][START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF]. Formally, this decomposition can be written:

T n = k,τ D(n) k,τ
where tensors D(n) k,τ are components of the irreducible decomposition, k denotes the order of the harmonic tensor embedded in D(n) and τ separates same order terms. This decomposition establishes an isomorphism between T n and a direct sum of harmonic tensor spaces H k [6]:

T n ∼ = k,τ H k,τ
but, as explained, in [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF] this decomposition may not be unique. Grouping together irreducible spaces of the same order, one obtains the O(3)-isotypic decomposition of a representation which is unique [START_REF] Kosmann-Schwarzbach | Groupes et symétries. Groupes finis, groupes et algébres de Lie, représentations[END_REF]:

T n ∼ = n k=0 α k H k
where α k is the multiplicity of the irreducible space H k in the decomposition and n is the order of the highest-order irreducible space of the decomposition. H k is the space of kth-order harmonic tensors, that is the space of totally symmetric, traceless tensors of order k. It is a vector space of dimension 2k + 1 in R 3 . The series {α k } is function of the order and the intrinsic symmetries of the tensor space. Various methods exist to compute this family [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF][START_REF] Zou | Orthogonal Irreducible Decompositions of Tensors of High Orders[END_REF][START_REF] Auffray | Démonstration du théoréme d'Hermann é partir de la méthode Forte-Vianello[END_REF].

Before closing this rather short introduction, it is important to note that in the decomposition of an even-order (resp. odd-order) tensor, odd-order (resp. even order) components are pseudo-tensors 2 , i.e change sign if the space orientation is reversed.

Dimension of G-invariant harmonic spaces

Let G be any subgroup of O(3). The set of tensors T ∈ T which are fixed by G Fix T (G) := {T ∈ T | g.T = T for all g ∈ G} , is called the fixed point set 3 . It is the biggest linear subspace of T that contains G-invariant tensors. Elements of Fix T (G) only defined in terms of G-invariance without any further constraint will be referred to as generic. Since non-generic elements constitute a null set they will not be considered here.

It is worth noting that the dimension of type II invariant subspaces is always 0. And for O(3)-subgroups of types I and III we have the following results 4 concerning harmonic tensor spaces.

Type I subgroups 5 dim Fix H k (Z p ) = 2 k p + 1 ; dim Fix H k (D p ) =    k p + 1 for k even k p for k odd dim Fix H k (T ) = 2 k 3 + k 2 -k + 1 ; dim Fix H k (O) = k 4 + k 3 + k 2 -k + 1 dim Fix H k (I) = k 5 + k 3 + k 2 -k + 1 Type III subgroups dim Fix H k (Z - 2p ) = 2 k + p 2p ; dim Fix H k (D v p ) =    k p if k = 2n k p + 1 if k = 2n + 1 dim Fix H k (D h 2p ) = k + p 2p ; dim Fix H k (SO(2) -) = 0 k = 2n 1 k = 2n + 1 dim Fix H k (O -) = k 3 - k 4
These elementary results, combined with the knowledge of the isotypic decomposition of a tensor space (3.1), allow to determine the dimension of any G-invariant tensor subspaces. Analytical expressions are constructed according to that procedure.

2. The related tensors are multiplied by the Levi-Civita symbol ijk .

3. As G as a action on the space T, there is an homeomorphism ψ from G to GL(T). Hence the notation g.T should be understand as a classical shortcut to the more rigorous one ψ(g).T.

4. More details can be found in [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF][START_REF] Auffray | Analytical expressions for anisotropic tensor dimension[END_REF]. 5. The formulas for type I subgroup were already provided in [START_REF] Auffray | Analytical expressions for anisotropic tensor dimension[END_REF]. They are summed-up here for the sake of completeness.

G-invariant tensor subspaces

Applying this process the following formulas are obtained:

Type I subgroups • Z p -invariance dim Fix T (Z p ) = 2 n k=0 α k k p + n k=0 α k
When p > k we obtain k p = 0 and so

β oth = n k=0
α k is the number of oriented transverse hemitropic coefficients. β oth is the dimension of a SO(2)-invariant tensor.

•D p -invariance dim Fix T (D p ) = n k=0 α k k p + n 2 k=0 α 2k
When p > k we obtain k p = 0 and so

β tti = n k=0
α k is the number of twisted transverse isotropic coefficients. β tti is the dimension of an O(2)-invariant tensor.

• T , O and I-invariance

dim Fix T (T ) = n k=0 α k (2 k 3 + k 2 -k + 1) ; dim Fix T (O) = n k=0 α k ( k 4 + k 3 + k 2 -k + 1) dim Fix T (I) = n k=0 α k ( k 5 + k 3 + k 2 -k + 1)
Type III subgroups

• Z - 2p -invariance dim Fix T (Z - 2p ) = 2 n k=0 α k k + p 2p
When p > k we obtain k+p 2p = 0 and therefore the tensor is null. In such a case, the tensor is obviously

O(3)-invariant. • D h 2p -invariance dim Fix T (D h 2p ) = n k=0 α k k + p 2p = 1 2 dim Fix T (Z - 2p )
When p > k we obtain k+p 2p = 0 and therefore the tensor is null. In such a case, the tensor is obviously

O(3)-invariant. • D v p -invariance dim Fix T (D v p ) = n k=0 α k k p + n-1 2 k=0 α 2k+1
When p > k we obtain k p = 0 and so

β th = n-1 2 k=0
α 2k+1 is the number of non-oriented transverse hemitropic coefficients. β th is the dimension of an O(2) --invariant tensor.

• O --invariance

dim Fix T (O -) = n k=0 α k k 3 - k 4

Physical results

In order to illustrate the practical interest of these formulas two examples will be considered: the thirdorder tensor of piezoelectricity and the fifth-order coupling tensor of Mindlin's strain gradient elasticity.

Piezoelectricity

Let us consider Piez the vector space of piezoelectricity tensors, its elements are symmetric under the permutation of their two first indices: P (ij)k , where (..) stands for the minor symmetry. It has been shown [START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Weller | Piezomagnetic tensors symmetries: an unifying tentative approach[END_REF] that this vector space is isomorphic to:

Piez ∼ = 2H 1 ⊕ H 2 ⊕ H 3
where the notation indicates a pseudo-tensor. And so Piez is defined by the following {α k } family:{0, 2, 1, 1}. As determined in [START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Weller | Piezomagnetic tensors symmetries: an unifying tentative approach[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF] the space of piezoelectric tensors can be divided into the following anisotropic systems:

[Piez] = {[1], [Z 2 ], [Z 3 ], [D v 2 ], [D v 3 ], [Z - 2 ], [Z - 4 ], [D 2 ], [D 3 ], [D h 4 ], [D h 6 ], [SO(2)], [O(2)], [O(2) -], [O -]}

Straightforward applications of our formula give

•Type I subgroups

[Piez] [1] [Z 2 ] [Z 3 ] [SO(2)] [D 2 ] [D 3 ] [O(2)] dim 18 8 6 4 3 2 1
•Type III subgroups

[Piez] [Z - 2 ] [Z - 4 ] [D v 2 ] [D v 3 ] [O -(2)] [D h 4 ] [D h 6 ] [O -] dim 10 4 5 4 3 2 1 1
where the number of coefficients for each physical symmetry class have been determined 6 . These results are obviously in agreement with the ones in the literature [START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Weller | Piezomagnetic tensors symmetries: an unifying tentative approach[END_REF][START_REF] Zou | Symmetry types of the piezoelectric tensor and their identification[END_REF].

6. A precise definition of symmetry classes can be found in the following references [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF][START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF].

Mindlin strain-gradient elasticity

Let us consider Ela M the vector space of the coupling tensors in Mindlin strain gradient elasticity. It is the vector space of fifth-order tensors endowed with the following index symmetries [START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF]: M (ij)(kl)m . It has been shown [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF] that this vector space decomposed as follows :

Ela M ∼ = H 0 ⊕ 6H 1 ⊕ 5H 2 ⊕ 5H 3 ⊕ 2H 4 ⊕ H 5
and so Ela M is defined by the following {α k } family:{1, 6, 5, 5, 2, 1}. As determined in [START_REF] Olive | Symmetry classes for odd-order tensors[END_REF] the space Ela M can be divided into the following anisotropic systems:

[Ela M ] = {[1], [Z 2 ], • • • , [Z 5 ], [D v 2 ], • • • , [D v 5 ], [Z - 2 ], • • • , [Z - 8 ], [D 2 ], • • • , [D 5 ], [D h 4 ], • • • , [D h 10 ], [SO(2)], [O(2)], [O(2) -], [T ], [O -], [O], [SO(3)], [O(3)]}
Therefore Ela M is divided into 28 symmetry classes 7 .This large number of symmetry classes has to be compared with the 8 symmetry classes of classical elasticity [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF], and the 17 symmetry classes of second order elasticity [START_REF] Auffray | Matrix representations for 3D strain-gradient elasticity[END_REF][START_REF] Olive | Symmetry classes for even-order tensors[END_REF]. Straightforward applications of our formula now give •Type I subgroups •Type III subgroups

[Ela M ] [1] [Z 2 ] [Z 3 ] [Z 4 ] [Z 5 ] [SO(2)] [D 2 ] [D 3 ] [D 4 ] [D 5 ] [O(2)] [T ] [O] [SO (3)
[Ela M ] [Z - 2 ] [Z - 4 ] [Z - 6 ] [Z - 8 ] [D v 2 ] [D v 3 ] [D v 4 ] [D v 5 ] [O -(2)] [D h 4 ] [D h 6 ] [D h 8 ] [D h 10 ] [O -] dim 56 26

Conclusion

In this note, analytical formulas giving the dimension of a subspace left fixed under O(3)-subgroups action have been provided and applied to the symmetry classes of some physical tensor spaces. To be applied, the only things that has to be known are the O(3)-isotypic decomposition of the studied tensor space, and the symmetry classes of the tensors spaces. Using the method proposed in [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF][START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF], this decomposition is easily obtained, and the determination of the symmetry classes is completely solved in [?,23]. We believe that these simple formulas can be of great help to develop, for example, higher-order constitutive laws [START_REF] Dell'isola | Generalized Hooke's law for isotropic second gradient materials[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF] and to design micro-macro homogenization procedure for anisotropic materials [START_REF] Trinh | Evaluation of generalized continuum substitution models for heterogeneous materials[END_REF].

  }, which are O(2)-closed subgroups; Exceptional groups : {T , O, I, SO(3)}, which are the rotation groups of chiral Platonic polyhedrons completed by the rotation group of the sphere.

7 .

 7 Or 29, if the class [O(3)] of null tensors is taken into account. But this point is only a question of convention O(3) type I closed-subgroups Hermann-Maugin Schonflies Group

For a given group G ∈ O(3), by generic G-invariant tensors we mean tensors that only satisfy G-invariance and no other constraint. This is the case of almost all G-invariant tensors.


Adrien Coulet

Florent Domenach
email: domenach.f@unic.ac.cy

Mehdi Kaytoue
email: mkaytoue@liris.cnrs.fr

Amedeo Napoli
email: napoli@loria.fr

Using pattern structures for analyzing ontology-based annotations of biomedical data

Annotating data with concepts of an ontology is a common practice in the biomedical domain. Resulting annotations, i.e., data-concept relationships, are useful for data integration whereas the reference ontology can guide the analysis of integrated data. Then the analysis of annotations can provide relevant knowledge units to consider for extracting and understanding possible correlations between data. Formal Concept Analysis (FCA) which builds from a binary context a concept lattice can be used for such a knowledge discovery task. However annotated biomedical data are usually not binary and a scaling procedure for using FCA is required as a preprocessing, leading to problems of expressivity, ranging from loss of information to the generation of a large number of additional binary attributes. By contrast, pattern structures offer a general FCA-based framework for building a concept lattice from complex data, e.g., a set of objects with partially ordered descriptions. In this paper, we show how to instantiate this general framework when descriptions are ordered by an ontology. We illustrate our approach with the analysis of annotations of drug related documents, and we show the capabilities of the approach for knowledge discovery.

Introduction

Annotating data resources with the concepts of an ontology is a common practice in the biomedical domain. The resulting annotations are reified as links between data and concepts of a "reference ontology", and provide a support for data exchange, data integration and data analysis tasks [START_REF] Daniel | Biomedical ontologies : a functional perspective[END_REF]. Usually annotations can be built in three main ways, manually, automatically and semi-automatically. In manual annotation, links between data and concepts are provided by human domain experts. In automated annotation, specialized programs are parsing data for providing such links. In semi-automated annotation, specialized programs are suggesting links between data and concepts, that are subsequently validated by domain experts [START_REF] Pesquita | Semantic similarity in biomedical ontologies[END_REF].

In the following, we are interested in the analysis of annotations of several data resources from different biomedical domains, e.g. molecular biology and medicine, w.r.t. a reference ontology. Indeed, the annotation process plays a major role in linking these different biomedical domains and understanding their relations.

In this way, this is one objective of translational bioinformatics to analyze molecular biology data along with clinical data for discovering correlations between them [START_REF] Atul | Viewpoint paper : Translational bioinformatics : Coming of age[END_REF]. Then, hypotheses about molecular mechanisms can be proposed through the discovery of correlations between molecular data and clinical observations. Such correlations can be discovered thanks to the analysis of annotations that link both molecular and clinical data to ontology concepts.

Formal Concept Analysis (FCA) is a mathematical framework for data analysis [START_REF] Ganter | Formal Concept Analysis[END_REF], which is a candidate for our knowledge discovery task. However, some adaptations are required as annotations can be considered as complex data. Firstly, given a reference ontology, annotations are considered as pairs <document, set of concepts> and cannot be directly represented as a binary context. Secondly, the ontology that encompasses the concepts used in the annotation should also be taken into account in the analysis. In FCA, several approaches exist for dealing with complex data.

A first approach is based on scaling, which relies on the transformation of non-binary data into binary data. Several types of scaling are known in FCA, e.g., nominal, ordinal, interordinal scalings [START_REF] Ganter | Formal Concept Analysis[END_REF]. But scaling leads to several problems such as an arbitrary transformation of data, a loss of information and a potential binary attribute flooding, forbidding a comprehensive visualization of the results (see for example experiments and discussion in [START_REF] Kaytoue | Revisiting numerical pattern mining with formal concept analysis[END_REF]).

Another approach is based on pattern structures that allows to directly analyze complex data whose descriptions are taken from a semi-lattice of descriptions [START_REF] Ganter | Pattern Structures and Their Projections[END_REF]. Descriptions may have various types, such as numerical intervals [START_REF] Kaytoue | Mining gene expression data with pattern structures in formal concept analysis[END_REF], set of attributes [START_REF] Ganter | Pattern Structures and Their Projections[END_REF] or graphs [START_REF] Kuznetsov | Learning closed sets of labeled graphs for chemical applications[END_REF]. However, a partial order on descriptions is required in pattern structures. This partial order is defined according to a similarity operator and an associated subsumption relation. Pattern structures allow for the reuse of standard FCA algorithms with slight modifications, for building the pattern concept lattice and all related operations. It can be noticed that the formalism of pattern structures has gained interest in the last years due to the need for FCA to analyze large volumes of complex data.

In this paper, we present an original approach to analyze annotations based on concepts lying in a reference ontology using the formalism of pattern structures. A first requirement for using pattern structures is to define descriptions of objects, then a similarity operation with its associated subsumption relation (thus a partial ordering on descriptions). In the present case, descriptions are based on concepts lying in a reference ontology. Accordingly, the ordering of concepts in the reference ontology is used to define an original similarity operator on object descriptions and the associated subsumption relation. This is -to the best of our knowledge-the first attempts to analyze data annotations thanks to a pattern structure. Moreover, this shows the potential of pattern structures as an effective formalism for dealing with real-world data. Actually, the resulting pattern concept lattice can be used for guiding a resource annotation process, and for completing annotations that are returned by an automatic annotation tool, that can be possibly wrong or incomplete. This is particularly valuable as the work of a domain expert for correcting and completing annotations is time consuming, especially when large corpora are considered.

From now, and for avoiding any confusion, we use the term "concept" for concept lying in ontologies (represented within Description Logics or DL) and "formal concept" or "pattern concept" for concepts in FCA and pattern structures.

The paper is organized as follows. Section 2 recalls fundamental definitions used in the paper. Section 3 presents our adaptation of pattern structures to ontology-based annotations. It introduces also a concrete example about biomedical data for illustrating the approach. Section 4 details the similarity and subsumption operations on descriptions, while Section 5 provides a discussion about the analysis of annotations of biomedical data using our approach.

2 Background definitions

Formal Concept Analysis

We recall here the standard FCA notations and we refer readers to [START_REF] Ganter | Formal Concept Analysis[END_REF] for details and proofs. A formal context (G, M, I) is defined as a set G of objects, a set M of attributes, and a binary relation I ⊆ G × M . (g, m) ∈ I means that "the object g is related with the attribute m through the relation I". Two derivation operators can be defined on sets of objects and sets of attributes as follows, ∀A ⊆ G, B ⊆ M :

A ′ = {m ∈ M ∶ ∀g ∈ A, (g, m) ∈ I} B ′ = {g ∈ G ∶ ∀m ∈ B, (g, m) ∈ I}
The two operators (⋅) ′ define a Galois connection between the power set of objects ℘(G) and the power set of attributes ℘(M ). A pair (A, B), A ⊆ G, B ⊆ M , is a formal concept iff A ′ = B and B ′ = A. A is called the extent and B the intent of the formal concept. The set of all formal concepts, ordered by inclusion of extents (or dually by inclusion of intents), i.e., (A 1 , B 1 ) ≤ (A 2 , B 2 ) iff A 1 ⊆ A 2 (or dually B 2 ⊆ B 1 ), forms a complete lattice [START_REF]Ordres et classification : Algèbre et combinatoire (tome II)[END_REF], called concept lattice.

Pattern structures

A pattern structure can be understood as a generalization of a formal context to analyze complex data [START_REF] Ganter | Pattern Structures and Their Projections[END_REF] : an object has a description lying in a semi-lattice where an "intersection" (or meet) is defined. This intersection allows for characterizing the similarity of two descriptions, i.e. what they do have in common.

Formally, let G be a set of objects, let (D, ⊓) be a meet-semi-lattice of object descriptions and let δ ∶ G → D be a mapping associating each object with its description. (G, (D, ⊓), δ) is called a pattern structure. gives rise to two derivation operators denoted by (⋅) ◻ :

A ◻ = ⊓ g∈A δ(g) f or A ⊆ G d ◻ = {g ∈ G d ⊑ δ(g)} f or d ∈ (D, ⊓).
These operators form a Galois connection between the power set of objects ℘(G) and (D, ⊓). Pattern concepts of (G, (D, ⊓), δ) are pairs of the form (A, d), A ⊆ G, d ∈ (D, ⊓), such that A ◻ = d and A = d ◻ . For a pattern concept (A, d), d is the pattern intent and is the common description to all objects in A, the pattern extent.

When partially ordered by

(A 1 , d 1 ) ≤ (A 2 , d 2 ) ⇔ A 1 ⊆ A 2 (⇔ d 2 ⊑ d 1 )
, the set of all pattern concepts forms a complete lattice called pattern concept lattice. The operator (⋅) ◻◻ is a closure operator and pattern intents are closed patterns. Pattern structures have been applied to numerical intervals [START_REF] Kaytoue | Mining gene expression data with pattern structures in formal concept analysis[END_REF] and to graphs [START_REF] Kuznetsov | Learning closed sets of labeled graphs for chemical applications[END_REF].

EL ontologies

Ontologies considered in this work are DL ontologies i.e., are based on a set of concepts, relations and individuals represented within Description Logic (DL) [START_REF] Baader | The Description Logic Handbook : Theory, Implementation, and Applications[END_REF]. Concepts can be either atomic or defined. In the first case, their description is reduced to a label and in the second case their description is a DL axiom that includes constructors such as conjunction and existential quantification.

The EL DL allows for conjunction (∧) and existential restriction (∃r.c) 1 in the definitions of concepts [START_REF] Baader | Pushing the el envelope[END_REF]. This simple DL is sufficient for our purpose, together with transitive roles and general concept inclusion axioms i.e., axioms of the form C ⩽ D where C, D can be either atomic or defined concepts. Moreover, the least common subsumer (lcs) of two concepts in EL always exists and can be computed in polynomial time, provided that there is no cycle in concept definitions, i.e., the definition of a concept c i does not make reference to c i itself [START_REF] Baader | Computing least common subsumers in description logics with existential restrictions[END_REF].

For avoiding any confusion and making a clear distinction between the DL formalism and the pattern structure formalism, we use the classical logical notations for the EL DL, thus ∧ for conjunction and ⩽ for subsumption, while we keep ⊓ for the similarity operator and ⊑ for the subsumption relation in pattern structures.

In the following, we consider a reference ontology denoted by O based on the EL DL. O is composed of :

-C(O) denotes a set of concepts and R(O) denotes a set of binary relations,

1. In addition we used a different operator to distinguish the DL subsumption (C ⩽ D) from the partial ordering on pattern concepts ((

A 1 , d 1 ) ≤ (A 2 , d 2 )) described in 2.2.
-concepts c i in C(O) are partially ordered thanks to a subsumption relation ⩽, where c 1 ⩽ c 2 means that concept c 1 is a sub-concept of c 2 and that every individual that is an instance of c 1 is an instance of c 2 , -A is a set of axioms that describe defined concepts.

3 Problem statement

The UMLS Semantic Network and semantic types

The UMLS (Unified Medical Language System) is composed of two main components : a set of ontologies of the biomedical domain (such as SNOMED CT, ICD-10, MeSH) and the UMLS Semantic Network [START_REF] Bodenreider | The Unified Medical Language System (UMLS) : integrating biomedical terminology[END_REF]. For sake of simplicity, we use a single data resource, DrugBank2 [START_REF] Knox | DrugBank 3.0 : a comprehensive resource for 'omics' research on drugs[END_REF] and a single ontology, the NCI (National Cancer Institute) Thesaurus [START_REF] Sioutos | NCI Thesaurus : A semantic model integrating cancer-related clinical and molecular information[END_REF], which belongs to the UMLS. Thus annotations that illustrate our study rely on links between DrugBank and the NCI Thesaurus.

The UMLS Semantic Network provides a set of broad subject categories, or semantic types, that is used as a high level classification for concepts of UMLS ontologies [START_REF] Mccray | An upper level ontology for the biomedical domain[END_REF]. An overview of the 133 semantic types is available at http://www.nlm.nih.gov/research/ umls/META3_current_semantic_types.html.

Semantic types are organized as a tree denoted hereafter as ST tree . For example, some semantic types are more general than others such as "Organism", which is more general than "Human" or "Anatomical Structure", which is more general than "Tissue".

Every concept of a UMLS ontology is mapped to one or more semantic types (i.e., to a non-empty set of semantic types). In addition, the hierarchy of ST tree can be used to map a concept c 1 to the set of semantic types that are ancestors of the semantic types of c 1 . For example, if the concept c 1 has for semantic type "Disease or Syndrome", it can be mapped to "Pathologic Function" and "Biologic Function" too (as the laters are ancestors of the former in ST tree ). Accordingly, we are using the hierarchy ST tree to dispose of the full set of semantic types that can be mapped to each concept. Figure 1 illustrates the mappings of some concepts of the NCI Thesaurus with their semantic types.

In our approach, a selection of semantic types chosen by the analyst will be used as upper level classes for concepts annotating biomedical documents.

Building a pattern structure for biomedical annotations

In this work, we are interested in the discovery of associations between sets of concepts annotating biomedical documents. This knowledge discovery method should take into account domain knowledge, i.e., the NCI Thesaurus and semantic types. For example, an expert may be interested in a drug-disease association, e.g., Antibiotic-Inflammation, checking whether the association is frequent and searching for a potential associated molecular mechanism.

For analyzing annotations it may be worth to distinguish concepts thanks to domains of interests (kinds of points of view). For example, a domain expert may group concepts according to their membership to distinct portions of an ontology to separate concepts about diseases from concepts about drugs. Accordingly, we consider in this work that the domain expert defines a set of dimensions ST = {st 1 , st 2 , ..., st k } where each st i is a semantic type. Then a biomedical document will be annotated w.r.t. ST dimensions. More precisely, given a biomedical document g, the annotation of g w.r.t. the reference ontology O and ST dimensions is a pair (g, ⟨ST 1 (g), ST 2 (g), ..., ST k (g)⟩) where ST i (g) is the set of concepts annotating g for the dimension st i of ST (possibly some of the ST i (g) can be empty).

For example, let us consider the document DB01082 (gathering data about Streptomycin) in the DrugBank database. Figure 2 shows this document and an anno-tation relating three concepts of the NCI Thesaurus (here the reference ontology O). Moreover, let us consider ST dimensions as ST = {Disease or Syndrome, Bacterium, Molecular Function, Chemical}. Then the annotation of DB01082 can be read as :

(DB01082, ⟨{T uberculosis}, {}, {P rotein Synthesis},

{Streptomycin}⟩)

Now we have everything for defining the pattern structure (G, (D, ⊓), δ) for analyzing annotations of biomedical documents :

-G = {g 1 , g 2 , ..., g n } is a set of annotated biomedical documents ; -O is the reference ontology, i.e., the NCI Thesaurus ; -ST = {st 1 , st 2 , ..., st k } is a subset of semantic types of the UMLS Semantic Network that defines the dimensions of the annotation vector ; -D = P(st 1 ) × P(st 2 ) × ... × P(st k ) where P(st i )

is the power set of the set of concepts of semantic type st i . As a product of complete lattices, D is also a complete lattice (and thus a semilattice). Elements of D are named hereafter ontological patterns ; δ ∶ G → D is a mapping associating a document g i ∈ G with a description in D or more precisely a vector in D,

δ(g i ) = ⟨ST 1 (g i ), ST 2 (g i ), ..., ST k (g i )⟩
where ST j (g i ) is the set of concepts of semantic type st j annotating g i . Table 1 gives an example of this pattern structure. The fourth row of the table shows the annotation of the document DB01082 (about Streptomycin). The different columns are filled with the concepts annotating DB01082 w.r.t. the semantic type provided in the header of each column. Now, it remains to define the similarity operation ⊓ between two descriptions δ(g 1 ) and δ(g 2 ) :

δ(g 1 ) = ⟨ST 1 (g 1 ), ST 2 (g 1 ), ..., ST k (g 1 )⟩ δ(g 2 ) = ⟨ST 1 (g 2 ), ST 2 (g 2 ), ..., ST k (g 2 )⟩ δ(g 1 ) ⊓ δ(g 2 ) = ⟨ST 1 (g 1 ) ⊓ ST 1 (g 2 ), ST 2 (g 1 ) ⊓ ST 2 (g 2 ), ..., ST k (g 1 ) ⊓ ST k (g 2 )⟩ where ST 1 (g 1 ) ⊓ ST 1 (g 2 ) is a light notation for ⟨ST 1 (g 1 )⟩ ⊓ ⟨ST 1 (g 2 )⟩. ST 1 (g 1 ) ⊓ ST 1 (g 2 ) is the convex hull in O of all concepts in ST 1 (g 1 ) and ST 1 (g 2 ).
The definition of the convex hull is made precise in the next section.

The similarity between descriptions

Given an ontology O, and two concepts c 1 and c 2 , the least common subsumer, denoted by lcs({c 1 , c 2 }), is the most specific concept subsuming both c 1 and c 2 w.r.t. the ontology O. Here O is an EL ontology where no cycle appears in concept definitions, thus the lcs of two concepts of O always exists [START_REF] Baader | Computing least common subsumers in description logics with existential restrictions[END_REF]. Indeed, the existence of the lcs is guaranteed as soon as O has a join semi-lattice structure. The lcs operation can be defined (recursively) for a set of concepts C n = {c 1 , c 2 , ..., c n } as follows :

∀n ∈ N, lcs(C n ) = lcs({lcs(C n-1 ), c n })
For example, the lcs of Streptomycin and Antif ungal Agent is Anti -Inf ective Agent (see Figure 2).

The lcs itself could be used to define a similarity operation between two descriptions. But, an objective here is to complete annotations of documents as much as possible. Thus, the convex hull operation appears to be a better similarity operation. Moreover, if one concept was missed by the annotation process but is available in the ontology, it can be retrieved within the convex hull of the initial set of annotating concepts.

The convex hull of the set of concepts {c 1 , c 2 }, denoted by CVX({c 1 , c 2 }), is defined as a set of concepts {x 1 , x 2 , ..., x n } verifying :

-

x i ⩽ lcs({c 1 , c 2 }), -(x i ⩾ c 1 and x i ∧ c 1 ≡ c 1 ) or (x i ⩾ c 2 and x i ∧ c 2 ≡ c 2 ), -x i ≢ ⊺ For example, CVX(Streptomycin, Antif ungal Agent) = {Aminoglycoside Antibiotic, Anti -Inf ective Agent, Antibiotic, Antif ungal Agent, Streptomycin}.
As the lcs operation, the convex hull operation can be generalized (recursively) to a set of concepts C p = {c 1 , c 2 , ..., c p } : We use the expression "convex hull" by analogy with the Euclidean geometry. In Euclidean geometry, a convex hull of a set of points is the minimal convex set that can be formed by these points. In our case, the convex hull of a set of concepts is the minimal set of concepts including the initial concepts, their least common subsumer and all concepts in between.

∀p ∈ N, CVX(C p ) = CVX({CVX(C p-1 ), c p })
The similarity operation on descriptions applies to two vectors having the same dimensions and returns a vector where the components are filled with the convex hull of the union of the two initial sets of concepts. Formally we have :

δ(g 1 ) = ⟨ST 1 (g 1 ), ST 2 (g 1 ), ..., ST k (g 1 )⟩ δ(g 2 ) = ⟨ST 1 (g 2 ), ST 2 (g 2 ), ..., ST k (g 2 )⟩ δ(g 1 ) ⊓ δ(g 2 ) = ⟨ST 1 (g 1 ) ⊓ ST 1 (g 2 ), ST 2 (g 1 ) ⊓ ST 2 (g 2 ), ..., ST k (g 1 ) ⊓ ST k (g 2 )⟩
where

ST i (g 1 ) ⊓ ST i (g 2 ) = CVX(ST i (g 1 ) ∪ ST i (g 2 )).
It can be noticed that the definition of the similarity operation on concepts can be likened to the the definition of the similarity operation for numerical intervals as the convex hull of two intervals (see for example [START_REF] Kaytoue | Mining gene expression data with pattern structures in formal concept analysis[END_REF]). Moreover, similarly as for intervals we have the following property :

δ(g 1 ) ⊓ δ(g 2 ) = δ(g 1 ) iff δ(g 1 ) ⊑ δ(g 2 )
As an illustration let us consider the two objects "Drug1" and "DB01082" and their descriptions δ(Drug1) and δ(DB01082) given in the The meet semi-lattice of pattern elements (actually of convex hulls) defined by the similarity operation is given in Figure 3. This semi-lattice is associated with the context of Table 1 and the order defined by the NCI Thesaurus given in Figure 2.

Dually, it is also possible to define a join operation on descriptions, making (D, ⊓, ⊔) a complete lattice. This operation is not necessary for the definition of pattern structures but exists in our case because of the property of D, the space of descriptions. The join of two descriptions δ(g 1 ) and δ(g 2 ) is defined as follows :

δ(g 1 ) ⊔ δ(g 2 ) = ⟨ST 1 (g 1 ) ⊔ ST 1 (g 2 ), ST 2 (g 1 ) ⊔ ST 2 (g 2 ), ..., ST k (g 1 ) ⊔ ST k (g 2 )⟩
where

ST i (g 1 ) ⊔ ST i (g 2 ) = CVX(ST i (g 1 )) ∩ CVX(ST i (g 2 )).
Actually, the result of the join operation is the set of common concepts in the two convex hulls of ST i (g 1 ) and ST i (g 2 ).

For example, the join of the descriptions of "Drug1" and "DB01082" is :

δ(Drug1) ⊔ δ(DB01082) = ⟨{T uberculosis},
{P rotein Synthesis}, {}⟩. 1 and the NCI Thesaurus. To enlighten the semi-lattice, we used abbreviations that are clarified in the upper right frame.

The intersection of two convex hulls may be empty as shown in the above example. However, it can be noticed that even if δ(g 1 ) and δ(g 2 ) may have no common element, they can still have a join as illustrates the following example. Suppose that we have only one dimension and let us consider the reference ontology in Figure 2 :

δ(g 1 ) = ⟨{Bacterial Inf ection, T uberculosis}⟩ δ(g 2 ) = ⟨{M ycobacterial Inf ection}⟩.
Actually, the results of the meet and join operations on these two descriptions are :

δ(g 1 )⊓δ(g 2 ) = ⟨{Bacterial Inf ection, M ycobacterial Inf ection, T uberculosis}⟩ δ(g 1 ) ⊔ δ(g 2 ) = ⟨{M ycobacterial Inf ection}⟩.
In addition, we remark that we do not have δ(g 1 ) ⊓ δ(g 2 ) = δ(g 1 ) as δ(g 1 ) is not a convex hull and thus we do not have either δ(g 1 ) ⊑ δ(g 2 ).

Computing Pattern Structures with Close-ByOne

In FCA, an efficient way of computing closed formal concepts that are the basic bricks of concept lattices is the algorithm CloseByOne [START_REF] Kuznetsov | A fast algorithm for computing all intersections of objects in a finite semilattice[END_REF][START_REF] Outrata | Fast algorithm for computing fixpoints of galois connections induced by object-attribute relational data[END_REF]. To adapt Close-ByOne to the general case of pattern structures, one has to replace the original Galois connection, usually denoted by (⋅) ′ , with the derivation operator denoted by (⋅) ◻ . Below, we give the basic pseudo-code of the algorithm CloseByOne (Algorithms 1 and 2) for computing patterns. In addition to the new derivation operator, one must replace the intersection of standard FCA with the similarity operation on patterns (⊓, line 5 of Algorithm 2) that is adapted to the nature of patterns. This adaptation of CloseByOne does not affect termination, correctness and complexity of the algorithm.

A simple implementation of Algorithms 1 and 2 is proposed at github.com/coulet/ OntologyPatternIcfca/.

Alg. 1 CloseByOne.

1: L = ∅ 2: for each g ∈ G 3:

process({g}, g, (g ◻◻ , g ◻ )) 4: L is the concept set.

Z = C ∪ {f } Y = D ⊓ {f ◻ } 6: X = Y ◻ process(Z, f, (X, Y )) 8: end if data
We illustrate our approach with the analysis of annotations of DrugBank documents with the ontology "NCI Thesaurus". These annotations are provided by the NCBO (National Center for Biomedical Ontology) Resource Index presented hereafter.

A repository of annotations : The NCBO Resource Index

The NCBO Resource Index is a repository of annotations automatically populated by a Natural Language Processing tool [START_REF] Jonquet | NCBO Resource Index : Ontology-based search and mining of biomedical resources[END_REF]. This tool parses the textual content of several biomedical databases (e.g., Drug-Bank, OMIM, ClinicalTrial.gov) searching for occurrences of terms refering to concepts of ontologies. When the name of a concept c i is found in a document g i , an annotation i.e., a pair (g i , c i ), is created and stored. On December 18th, 2012, the NCBO Resource Index contained annotations for 34 databases with concepts of 280 ontologies of the BioPortal [START_REF] Whetzel | Bio-Portal : enhanced functionality via new web services from the national center for biomedical ontology to access and use ontologies in software applications[END_REF]. The Resource Index can be queried either by a Web user interface 3 or by a REST Web service 4 . We used the latter to build sets of annotations.

DrugBank annotations with the NCI Thesaurus

DrugBank is a publicly available database that contains data about drugs, their indications and their molecular targets. The database is organized into documents, or entries, where each document compiles data about one drug. Data in DrugBank are for the main part made of texts is natural language. Annotations considered in the following relate Drug-Bank documents and concepts of the "NCI Thesaurus" ontology. The NCI Thesaurus is a broad domain ontology and consequently its annotations may concern either clinics or molecular biology data that can be conjointly explored in translational bioinformatics. Moreover, the NCI Thesaurus is an EL ontology, thus a lcs always exists and its processing is tractable. We used the version 12.04 of the NCI Thesaurus encoded in OWL and available on the NCBO Bioportal 5 . 3 and the corresponding pattern concept lattice is given in Figure 4. Both sets of formal concepts in the semilattice and in the pattern concept lattice were obtained thanks to the adapted implementation of CloseByOne (see subsection 3.4). Now we propose an analysis of the resulting concept lattice shown in Figure 4. Consider that one of our objectives is to repair and complete the annotations associated with biomedical documents. The top formal concept in the lattice has the "largest extent", i.e., the set of all the objects, and the "smallest intent", actually the largest convex hull for the annotations.

Let us consider the two formal concepts in the upper left part of the concept lattice, the first called here c #15 has an extent containing "drug1" and "drug5" and the second called here c #5 has an extent containing only "drug5". The Chemical semantic type of both concepts is {Anti -Inf ective Agent, Antibiotic, Antif ungal Agent}. The Disease or Syndrome dimension ("DoS") in both concepts is {Bacterial Inf ection, M ycobacterial Inf ection, T uberculosis} as in the top concept. However, the Molecular Function dimension ("MF") is the same for the top concept and c #15 , i.e., {P rotein Synthesis} ; while it is redefined and empty in c #5 . This can be interpreted as follows :

-The value of Chemical in both c #15 and c #5 is completed (as a convex hull) with Anti-Infective Agent and is the correct annotation to be associated to documents "drug1" and "drug5" for the Chemical dimension. This shows how the final pattern concept lattice can effectively complete the original annotation process (especially when this process is automated). -The same remark applies to the Disease or Syndrome dimension, which is also completed (as a convex hull). The concept lattice provides once again the complete annotation for both concepts c #15 and c #5 . Thus, even on this small and toy example, it is possible to understand and verify the usefulness and potential of the approach : the resulting pattern concept lattice yielded by the "ontological pattern structure" provides the means for completing the initial annotations in a way that respects the reference ontology.

Finally, we experimented the pattern structure ap- The resulting concept lattice is rather large and the analysis of formal concepts with a domain expert is in progress. We think that the results of the analysis will be in accordance with the analysis presented just above for the toy example.

Conclusion and Perspectives

Pattern structures provide an original and effective approach within FCA to analyze complex data such as ontology-based annotations of biomedical documents. In this paper, we propose a framework based on pattern structures for dealing with annotations which are made with concepts represented within an EL ontology. Then we propose a pattern structure providing a classification of biomedical documents according to their annotations and the semantic types of the concepts within the annotations. The resulting concept lattice can be used for analyzing and completing the original annotations.

This work shows that pattern structures are an ef-fective means for dealing with real-world and complex data. In the present case, more experiments remain to be done as well as a thorough study of the various pattern structures that can be associated to an annotation process depending on one or several ontologies.

  Elements of D are called descriptions or patterns and are ordered by a subsumption relation ⊑ such as ∀c, d ∈ D, c ⊑ d ⇐⇒ c⊓d = c. A pattern structure (G, (D, ⊓), δ)

Figure 1 -

 1 Figure 1 -Detail of the NCI Thesaurus with associated semantic types from the UMLS. Nodes are concepts of the ontology, arrows represent subsumption relationships (⩽). Doted lines map each concept to its semantic type as defined in the UMLS Semantic Network.

Figure 2 -

 2 Figure 2 -(a) The left part of the Figure shows the NCI Thesaurus ontology ; (b) the right part is an excerpt of the document DB01082 of DrugBank related to the Streptomycin drug. Bold arrows connecting (a) and (b) represent the annotation of DB01082.

Figure 3 -

 3 Figure 3 -The meet semi-lattice of convex hulls associated with the context represented in Table1and the NCI Thesaurus. To enlighten the semi-lattice, we used abbreviations that are clarified in the upper right frame.

Alg. 2

 2 process(A, g, (C, D)) with C = A ◻◻ and D = A ◻ and < the lexical order on object names. if {h h ∈ C A and h < g} = ∅ then 2: L = L ∪ {(C, D)} for each f ∈ {h h ∈ G C and g < h} 4:

Figure 2 (

 2 b) presents the document of DrugBank in concern with Streptomycin.

Figure 4 -

 4 Figure4-The pattern concept lattice corresponding to the pattern structure given in Table1and on the NCI Thesaurus. The top concept has the intent with the larger descriptions and consequently its extent includes all the documents (objects). Traversing the lattice downward, the concepts present more specialized extents and more general intents w.r.t. the subsumption relation on descriptions "Ch", "DoS" and "MF" are respectively abbreviations for the semantic types Chemical, Disease or Syndrome and Molecular Function.

Table 1 -

 1 A pattern structure where objects are DrugBank documents and attributes are semantic types. Each document is annotated with a set of concepts of the NCI Thesaurus (the reference ontology) having distinct semantic types. The document DB01082 of DrugBank is annotated with three concepts, including the concept Tuberculosis of semantic type Disease or Syndrome.

	P P P P P G ST	Disease or Syndrome	Bacterium	Molecular Function	Chemical
	Drug1	{Tuberculosis, Bacterial Infection}	{}	{Protein Synthesis}	{Antibiotic, Antifungal Agent}
	Drug2	{Bacterial Infection}	{}	{Protein Synthesis}	{}
	Drug3	{Tuberculosis, Bacterial Infection}	{}	{}	{Anti-Infective Agent}
	DB01082	{Tuberculosis}	{}	{Protein Synthesis}	{Streptomycin}
	Drug5	{Tuberculosis, Bacterial Infection}	{}	{}	{Antibiotic, Antifungal Agent}

Table 1

 1

	. Their

  We propose in Table1a context including annotations of five DrugBank documents based on concepts to the NCI Thesaurus. Concepts may have four distinct semantic types ( ST = 4) : Disease or Syndrome, Bacterium, Molecular Function and Chemical. The meet-semi-lattice of patterns associated with such annotations is depicted in Figure

			4.3 Interpretation
	3. Available	at	http://bioportal.bioontology.org/
	resource_index
	4. Documented at http://www.bioontology.org/wiki/
	index.php/Resource_Index_REST_Web_Service_User_Guide
	5. NCI Thesaurus 12.04 : bioportal.bioontology.org/
	ontologies/1032

Publicly available at http://www.drugbank.ca/

Acknowledgement

This work was supported in part by the funding agency Campus France (Zenon PHC project 24855NG) and by the Research Promotion Foundation of Cyprus (project KD4CD DIAKRATIKES/KY-GA/0310).


D Bichara
email: derdei.bichara@inria.fr

N Cozic

A Iggidr

Epi Masaie

On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients

Keywords: Mathematics Subject Classification: 92C60, 92D30, 93B07, 93C15, 93C41 Intra-host epidemic models, observers, detectability, age-structured models

The aim of this paper is to give a method for the estimation of total parasite burden of the patient and the rate of infection in a malaria's intra-host model by using control theory tools. More precisely, we use an auxiliary system, called observer or estimator, whose solutions tend exponentially to those of the original model. This observer uses only the available measurable data, namely, the values of peripheral infected erythrocytes. It provides estimates of the sequestered infected erythrocytes, that cannot be measured by clinical methods. Therefore this method allows to estimate the total parasite burden within a malaria patient. Moreover, our constructed observer does not use the uncertain infection rate parameter β. In fact, we derive a simple method to estimate this parameter β. We apply this estimation method using real data that have been collected when malaria was used as therapy for neurosyphilis by the US Public Health Service.

Introduction

Malaria is a disease that causes at least one million deaths around the world each year, with ninety per cent among African children. the most dangerous type of malaria is caused by the most virulent species of the Plasmodium parasite: Plasmodium falciparum. Intra-host models describe the dynamics of the interaction of parasites (viruses, bacteria, protozoans, ...) within the host. These models have been used to understand the population dynamics and evolution of Plasmodium falciparum in the host. An abundant literature has been dedicated to malaria's intra-host models, one can see for instance [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF][START_REF] Gravenor | Reply to : Models for the in-host dynamics of malaria revisited : errors in some basic models lead to large over-estimates of growth rates[END_REF][START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF][START_REF] Mason | Blood-stage dynamics and clinical implications of mixed plasmodium vivax-plasmodium falciparum infections[END_REF][START_REF] D P Mason | The blood-stage dynamics of mixed plasmodium malariae-plasmodium falciparum infections[END_REF] and references therein. A review of some models has been given in [START_REF] Molineaux | Review of intra-host models of malaria[END_REF]. Most of these papers are, in the whole, related to the seminal Anderson, May and Gupta's model [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF] which is the 3 dimensional system:

       ẋ = Λ -µ x x -βxm, ẏ = βxm -µ y y, ṁ = rµ y y -µ m m -βxm, (1)
where x is the concentration of uninfected erythrocytes in the blood, y is the concentration of infected erythrocytes, and m the concentration of free merozoites. Parameters µ x , µ y and µ m are the death rates of the uninfected erythrocytes, infected parasites and free merozoites respectively. The parameter β is the contact rate between erythrocytes and merozoites. Uninfected blood cells are recruited at a constant rate Λ from the bone marrow and have a natural life-expectancy of 1/µ x × days. Death of an infected erythrocyte results in the release of an average number of r merozoites. Free merozoites die or successfully invade a healthy erythrocyte. For the epidemiology of malaria, we refer to [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF][START_REF] Saul | Transmission dynamics of plasmodium falciparum[END_REF]. As we can see in [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF], the dynamics of the infected erythrocytes are described by one differential equation. However, experimental analysis has shown that the dynamics of Plasmodium falciparum might change according to its "age", i.e., the stage of its life cycle. Hence, a model, where the dynamics of the healthy cells are coupled with an age-structured model for the infected cells and the dynamics of free merozoites, has been considered [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF][START_REF] Gravenor | Reply to : Models for the in-host dynamics of malaria revisited : errors in some basic models lead to large over-estimates of growth rates[END_REF]:

                       ẋ = Λ -µ x x -β xm, ẏ1 = β xm -(γ 1 + µ 1 ) y 1 , ẏ2 = γ 1 y 1 -(γ 2 + µ 2 ) y 2 , . . . ẏn = γ n-1 y n-1 -(γ n + µ n ) y n , ṁ = r γ n y n -µ m m -β xm. (2)
One of the characteristics of Plasmodium falciparum is sequestration. To explain, let us give a brief description of the Plasmodium life cycle. The cycle begins when the parasite enters the human body through the bite of an infected mosquito, after which it migrates to and multiplies within the liver. Then the free forms (merozoites) resulting from this multiplication are able to invade the red blood cells (erythrocytes). Erythrocytes which are infected, mature during the erythrocytic cycle. At roughly the middle stage of trophozoite development (24 hours), molecules on the surface of infected erythrocytes link to receptors of endothelial cells. This bind has the effect of holding infected erythrocytes within vessels of organs (such as the brain), where they remain until the rupture of the erythrocyte and the release of merozoites. This period of attachment is called sequestration and during it, the infected erythrocytes are not detectable in the blood flow, they are "sequestered". Also it is widely accepted that antimalarial drugs act preferentially on different stage of parasite development [START_REF] M B Gravenor | Estimating sequestered parasite population dynamics in cerebral malaria[END_REF][START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF]. In practice, to know a patient's stage of infection, the total parasite concentration n i=1 y i in the bloodstream is needed. However, only the peripheral infected erythrocytes (young parasites y 1 + y 2 + . . . y k , for some k < n), also called circulating, can be observed (seen on peripheral blood smears) and the other ones (sequestered: y k+1 , . . . y n ) are hidden in some organs like brain and heart, and cannot be observed. There is no clinical method of measuring the sequestered infected cells directly. All parameters in (2) can be estimated by biological considerations, except the parameter β which is the Red Blood Cells (RBC)'s rate of infection. The rate of infection is generally an unknown in epidemiology models. The estimation of sequestered parasite population has been a challenge for the biologist and modeler, with many authors [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF][START_REF] M B Gravenor | Estimating sequestered parasite population dynamics in cerebral malaria[END_REF][START_REF] Ochola | Estimating sequestered parasite load in severe malaria patients using both host and parasite markers[END_REF] having studied this problem. In this paper, we propose a different and simple method to estimate the total parasite concentration from the measured circulating parasites and we show how the parameter β can be estimated. To this end, we use some tools from control theory. More precisely, we built an observer, i.e: an auxiliary dynamical system whose states tend exponentially to the states of the original system [START_REF] William | A retrospective examination of sporozoite-and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: effect on development of parasitologic and clinical immunity[END_REF]. Moreover, this auxiliary system does not depend on the badly known parameter β. The main idea here is to consider the infection term βxm as an unknown input and to use the theory of observers for systems with unknown inputs [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF][START_REF] Hou | Disturbance decoupled observer design: A unified viewpoint[END_REF] to provide the dynamical estimates.

Dynamical estimation of the hidden parasitized erythrocytes

The exact number of stages of parasitized erythrocytes is unknown but according to [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF] there are five main stages defined by simple morphology: young ring, old ring, trophozoite, early schizont and finally late schizont. So we suppose that the parasitized erythrocytes population within the host is divided in 5 different stages: y 1 , y 2 , y 3 , y 4 , y 5 . The two first stages correspond to the concentration of free circulating parasitized erythrocytes and the three last stages correspond to the sequestered ones. Let x be the concentration of healthy cells, and m the concentration of merozoites. The healthy cells x are produced by a constant recruitment Λ from the thymus and they become infected by an effective contact with a merozoite m. At the late stage of infected cells, the erythrocyte ruptures and releases r merozoites. The model that we consider has a seven dimensional state space, given by

                               ẋ = Λ -µ x x -β xm, ẏ1 = β xm -(γ 1 + µ 1 ) y 1 , ẏ2 = γ 1 y 1 -(γ 2 + µ 2 ) y 2 , . . . ẏ5 = γ 4 y 4 -(γ 5 + µ 5 ) y 5 , ṁ = r γ 5 y 5 -µ m m -β xm. (3)
The different parameters of this model are defined as follows:

• Λ : Recruitment of the healthy red blood cells (RBC).

• β : Rate of infection of RBC by merozoites.

• µ x : Natural death rate of healthy cells.

• µ i : Natural death rate of i-th stage of infected cells.

• γ i : Transition rate from i-th stage to (i + 1)-th stage of infected cells.

• r : Number of merozoites released by the late stage of infected cells.

• µ m : Natural death rate of merozoites.

As suggested in [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF], we suppose that we can measure the circulating parasitaemia, i.e: y 1 + y 2 and we want to find an estimate of the sequestered parasitaemia, y 3 + y 4 + y 5 . Here, we use some classical notations from control theory. Let z(t) be the state at time t of the system (3), i.e., z = (x, y 1 , . . . , y 5 , m) ∈ IR 7 . The measurable output of system (3) will be denoted by Y. Here, the measurable output corresponds to the concentration of the circulating parasitized erythrocytes, and hence we have Y(t) = y 1 (t) + y 2 (t). System (3) can then be written as:

ż = A z + β xm E + Λ e 1 , (4a)
Y = C z, (4b)
where z ∈ IR 7 and Y ∈ IR are the state vector and the measurable output, respectively. The matrices A, C, E and e 1 are defined as follows:

A =               -µ x 0 0 0 0 0 0 0 -µ 1 -γ 1 0 0 0 0 0 0 γ 1 -µ 2 -γ 2 0 0 0 0 0 0 γ 2 -µ 3 -γ 3 0 0 0 0 0 0 γ 3 -µ 4 -γ 4 0 0 0 0 0 0 γ 4 -µ 5 -γ 5 0 0 0 0 0 0 rγ 5 -µ m               , E =               -1 1 0 0 0 0 -1               , e 1 =               1 0 0 0 0 0 0              
, and C = (0, 1, 1, 0, 0, 0, 0).

The equation (4a) describes the dynamics of the system while equation (4b) gives information about what is measured.

The problem we are interested in is: how to use the dynamical model (4a) together with the information provided by the measurements (4b) in order to get estimates of the state variables that cannot be measured? A solution to this problem can be provided by what is called in Control Theory an observer or estimator. An observer is an auxiliary dynamical system which uses the information Y(t) provided by the system and which produces an estimate ẑ(t) of the state z(t) such that the estimation error ẑ(t) -z(t) tends exponentially fast to zero as time t goes to infinity. An observer for system (4a-4b) will be of the form

ẇ(t) = f ( ŵ(t), Y(t)), ẑ(t) = g(w(t), Y(t)). (5)
One has to find the functions f and g in such a way that the solutions ẑ(t) of ( 5) and the solutions z(t) of (4a) satisfy for all initial conditions (ẑ(0), z(0)):

ẑ(t) -z(t) ≤ exp(-λ t) ẑ(0) -z(0)
for some positive real number λ.

However, the construction of the observer (5) requires the knowledge of all the parameters of the model. This is not the case for our system since the parameter β is unknown. Therefore we are going to consider the term βxm appearing in the system (4a) as an unknown input and we shall use the notation:

d(t) = β x(t) m(t)
. With this, system (4a) can be considered as a linear system with unknown input d(t) and can be written:

ż = A z + Λ e 1 + d(t) E, Y = C z. (6)
Following the approach from [START_REF] Hou | Disturbance decoupled observer design: A unified viewpoint[END_REF], we shall construct an estimator (observer) for system [START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF].

The idea for the construction will become evident after the proof of Theorem 2.1 below.

We define the matrix Ā by

Ā = (I d -E C) A
where I d denotes the identity matrix. For our system, we have

Ā =                    -µ x -µ 1 -µ 2 -γ 2 0 0 0 0 0 -γ 1 µ 2 + γ 2 0 0 0 0 0 γ 1 -µ 2 -γ 2 0 0 0 0 0 0 γ 2 -µ 3 -γ 3 0 0 0 0 0 0 γ 3 -µ 4 -γ 4 0 0 0 0 0 0 γ 4 -µ 5 -γ 5 0 0 -µ 1 -µ 2 -γ 2 0 0 r γ 5 -µ m                    .
It can be shown that the pair ( Ā, C) is detectable which means that there exists a column matrix

L = (l 1 , l 2 , . . . , l 7 ) t ∈ IR 7
such that the matrix Ā -LC is Hurwitz, i.e., all its eigenvalues have a negative real part.

We can now give our main result. Now we have all the ingredients to derive a method to dynamically estimate the state z(t) of our system.

Theorem 2.1. An exponential observer for system (6) is given by

   ẇ(t) = ( Ā -LC) w(t) + L + ( Ā -LC)E Y(t) + Λ e 1 , ẑ(t) = w(t) + E Y(t). ( 7
)
This observer is inspired from [START_REF] Hou | Disturbance decoupled observer design: A unified viewpoint[END_REF].

Proof. We prove that the estimation error e(t) = ẑ(t) -z(t) converges exponentially towards zero. Indeed, we have

ė = ż - ż = ẇ + E Ẏ -Āz -E Ẏ -Λe 1 = ( Ā -LC)w + {(L + ( Ā -LC)E} Y + Λe 1 + E Ẏ -Āz -E Ẏ -Λe 1 = ( Ā -LC)w + {(L + ( Ā -LC)E} Y -Āz = ( Ā -LC)(ẑ -E Y) + {(L + ( Ā -LC)E} Y -Āz = ( Ā -LC)ẑ -( Ā -LC)E Y + L Y + ( Ā -LC)E Y -Āz = ( Ā -LC)ẑ + LCz -Āz = ( Ā -LC)(ẑ -z) = ( Ā -LC)e
This shows that e(t) → 0 with exponential convergence rate since all the eigenvalues of the matrix Ā -LC have negative real parts.

Remark 2.1. It is remarkable that the computation of the estimates ẑ(t) does not depend on the parameter β. Moreover, the estimation method proposed in Theorem 2.1 is valid for arbitrary number n of age-classes.

For our model with n = 5, we have

Ā -LC =                    -µ x -µ 1 -l 1 -µ 2 -γ 2 -l 1 0 0 0 0 0 -γ 1 -l 2 µ 2 + γ 2 -l 2 0 0 0 0 0 γ 1 -l 3 -µ 2 -γ 2 -l 3 0 0 0 0 0 -l 4 γ 2 -l 4 -µ 3 -γ 3 0 0 0 0 -l 5 -l 5 γ 3 -µ 4 -γ 4 0 0 0 -l 6 -l 6 0 γ 4 -µ 5 -γ 5 0 0 -µ 1 -l 7 -µ 2 -γ 2 -l 7 0 0 r • γ 5 -µ m                    The eigenvalues of Ā -LC are: -µ x , -(γ 1 + γ 2 + µ 2 ), -(µ 3 + γ 3 ), -(µ 4 + γ 4 ), -(µ 5 + γ 5 ),
-µ m , and -(l 2 + l 3 ). Therefore the matrix Ā -LC is stable provided that l 2 + l 3 > 0. Since the eigenvalues of Ā -LC do not depend on l 1 nor on l i for i = 4, ..7, we simply choose

l 1 = l 4 = l 5 = l 6 = l 7 = 0.
The eigenvalues -(l 2 + l 3 ) and -(γ 1 + γ 2 + µ 2 ) come from the 3 × 3 upper left block, and it is somehow a stroke of luck that they are so simple algebraically.

Therefore the dynamical estimator [START_REF] M B Gravenor | Estimating sequestered parasite population dynamics in cerebral malaria[END_REF] for n = 5 is given by

                                                                                         ẇ1 = Λ -µ x w 1 -µ 1 w 2 -(µ 2 + γ 2 ) w 3 + (µ x -µ 1 ) Y ẇ2 = -(γ 1 + l 2 ) w 2 + (µ 2 + γ 2 -l 2 ) w 3 -γ 1 Y ẇ3 = (γ 1 -l 3 ) w 2 -(µ 2 + γ 2 + l 3 ) w 3 + γ 1 Y ẇ4 = γ 2 w 3 -(µ 3 + γ 3 ) w 4 ẇ5 = γ 3 w 4 -(µ 4 + γ 4 ) w 5 ẇ6 = γ 4 w 5 -(µ 5 + γ 5 ) w 6 ẇ7 = -µ 1 w 2 -(µ 2 + γ 2 ) w 3 + r γ 5 w 6 -µ m w 7 + (µ m -µ 1 ) Y ẑ1 = w 1 -Y ẑ2 = w 2 + Y ẑ3 = w 3 ẑ4 = w 4 ẑ5 = w 5 ẑ6 = w 6 ẑ7 = w 7 -Y (8)
Thanks to this observer, one can compute easily w(t) and then ẑ(t) which tends exponentially fast to z(t). The estimates of respectively x(t), y i (t), m(t) are respectively x(t) = ẑ1 (t), ŷi (t) = ẑi+1 (t) for i = 1, . . . 5, and m(t) = ẑ7 (t).

Therefore one can get an estimate of the total parasite burden

A method for the estimation of the rate of infection β

Let us consider the dynamic of y 1 , we have ẏ1 = βxm -(γ 1 + µ 1 ) y 1 . By the variation of constants method, we can write:

y 1 (t) = y 1 (t 0 )e -(γ 1 +µ 1 ) (t-t 0 ) + β t t 0
x(s)m(s)e (γ 1 +µ 1 ) (s-t) ds By replacing x(t), y 1 (t) and m(t) by their estimates x(t), ŷ1 and m(t) provided by the estimator (8) , we have:

ŷ1 (t) = ŷ1 (t 0 )e -(γ 1 +µ 1 ) (t-t 0 ) + β t t 0
x(s) m(s)e (γ 1 +µ 1 ) (s-t) ds Then ŷ1 (t)e (γ 1 +µ 1 ) t -ŷ1 (t 0 )e (γ 1 +µ 1 ) t 0 = β t t 0

x(s) m(s)e (γ 1 +µ 1 ) s ds

Let t f be the time after which the measured peripheral parasitaemia Y becomes practically zero. We discretize the interval [t 0 , t f inal ] in [t i , t i+1 ] and we write the relation ( 9) for each interval [t i , t i+1 ] to obtain ŷ1 (t i+1 )e (γ 1 +µ 1 ) t i+1 -ŷ1 (t i )e (γ 1 +µ 1 ) t i = β

t i+1 t i
x(s) m(s)e (γ 1 +µ 1 ) s ds [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF] For each i, let

U i = ŷ1 (t i+1 )e (γ 1 +µ 1 ) t i+1 -ŷ1 (t i )e (γ 1 +µ 1 ) t i , and V i = t i+1 t i
x(s) m(s)e (γ 1 +µ 1 ) s ds.

By numerical integration, we can compute, for each i, U i and V i . Thus, [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF] can written as:

U = βV
where U and V are vectors of appropriate dimension with U i and V i as components of U and V respectively. Hence, by a linear regression, we can obtain β which is an estimate of β.

4 Illustration using malariatherapy data

Estimation of the numbers of sequestered infected erythrocytes

In this section we show how our method to estimate the total parasitaemia from the peripheral infected cells that are measured can be applied. By peripheral parasitaemia, we refer the density of parasites in the blood. The measured peripheral parasitaemia are taken from data collected when malaria was used as therapy for neurosyphilis by the US Public Health Service at the National Institutes of Health laboratories in Columbia, South Carolina and Milledgeville, Georgia [START_REF] William | A retrospective examination of sporozoite-and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: effect on development of parasitologic and clinical immunity[END_REF]. Patients were inoculated through mosquito bite or infected blood [START_REF] William | A retrospective examination of sporozoite-and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: effect on development of parasitologic and clinical immunity[END_REF][START_REF] Eichner | Genesis, sequestration and survival of plasmodium falciparum gametocytes: parameter estimates from fitting a model to malariatherapy data[END_REF]. Those data have been successfully used by many authors [START_REF] William | A retrospective examination of sporozoite-and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: effect on development of parasitologic and clinical immunity[END_REF][START_REF] Eichner | Genesis, sequestration and survival of plasmodium falciparum gametocytes: parameter estimates from fitting a model to malariatherapy data[END_REF][START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF][START_REF] Molineaux | Malaria therapy reinoculation data suggest individual variation of an innate immune response and independent acquisition of antiparasitic and antitoxic immunities[END_REF] for different purposes concerning the Plasmodium falciparum parasite. To apply our total parasitaemia estimation method, we choose data of four patients named S1204, S1050, G221, and G54. These patients have been studied in [START_REF] Eichner | Genesis, sequestration and survival of plasmodium falciparum gametocytes: parameter estimates from fitting a model to malariatherapy data[END_REF] for the following reasons: they do not have any of the following features:

• (i) treatment with drugs reported to affect gametocytes (chlorguanide, pyri-methamine, primaquine) or unknown drugs;

• (ii) super-inoculation with any malaria parasite in the course of the infection;

• (iii) parasitological status on the day before onset of curative treatment positive or uncertain;

• (iv) gametocyte counts exceeding 100 µl (otherwise a reliable estimation of parameters would have been impossible with the current model);

• (iv) inoculation of a rarely used strain.

Those particular patients are relevant to our study because those patients have not taken treatment with drugs reported to affect gametocytes [START_REF] Eichner | Genesis, sequestration and survival of plasmodium falciparum gametocytes: parameter estimates from fitting a model to malariatherapy data[END_REF] and then satisfy the assumption of low mortality rates for peripheral erythrocytes that we make here. This assumption has also been used in [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF]. It remains to pick the numerical values for different parameters. Among parameters in (3), some of them are known or at least widely accepted. However, the infection rate β is unknown or hardly known, so our method here allows us to study the system without explicitly using β. With many classes, the most difficult task is how to choose the transition rates [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF][START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF] with the condition that the sum of the stage waiting times (life-span of parasitized erythrocytes) is 2 days. So, in this paper we use the transition and death rates from [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF] for patients without fever, i.e: at 37 With these parameters, we will obtain

5 i=1 1 γ i + µ i = 1.
9101065 days, which is close to 2 days.

In this paper, the unit of volume is micro-liter (µl) and the unit of time is day. The life span of healthy erythrocytes is approximately 120 days [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF][START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF], so µ x = 1 120 day -1 .

For the constant recruitment, we proceed as in [START_REF] Philip | Age-structured red blood cell susceptibility and the dynamics of malaria infections[END_REF], where the authors fixed the constant rate of erythrocytes' production at 5 × 10 6 120 cells µl -1 day -1 . This quantity is obtained from the parasite-free equilibrium state: 0 = Λ -µ x x 0 where x 0 is the fixed concentration of RBC in adult males which is around 5 × 10 6 RBCs per µl.

Merozoite's life-span ( 1 µ m

) is, according to [START_REF] R M Anderson | Non-linear phenomena in host-parasite interactions[END_REF], about 20 minutes, so µ m = 72 day -1 .

The number of merozoites released in the blood is between 8-32. So, we take here r = 16.

Recall that the choice l 2 and/or l 3 positive make the the matrix Ā -LC asymptotically stable. More precisely the spectrum of the matrix Ā -LC is:

Sp( Ā -LC) = {-l 2 -l 3 ; -µ x ; -µ m ; -µ 5 -γ 5 ; -µ 4 -γ 4 ; -µ 3 -γ 3 ; -γ 1 -µ 2 -γ 2 }
For the simulations we take l 2 = l 3 = 5. Since all the eigenvalues of the matrix Ā -LC are negative, the estimation error will tend exponentially to zero. For a given precision ǫ > 0, it is possible to compute a minimal time t ǫ satisfying | ŷi (t) -y i (t)| < ǫ ẑ(0) -z(0) for all t > t ǫ .

In practice, to determine t ǫ , it is sufficient to run the estimator [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF] with different initial conditions and then the estimates become reliable from the time when the various curves merge. An illustration for the total parasite burden is given in Figure 1. For the patient S1204, t ǫ ≈ 5 days.

Hereafter, we present the estimation results for the patient S1204 obtained by the estimator (8) using the malariatherapy data. The total parasitaemia estimates for the other patients are given in Figures 6,7 The estimates of sequestered and total parasite population per mico-liter in patient S1204 are given in Table 1.

Time in days

Partial synchrony

One of the most important feature of malaria is the synchronous development of the parasite within the host, i.e. all of the infected RBCs (rings, trophozoites, shizontes) are about the same during the development of the cycle. This synchrony leads to simultaneous eruption of all mature schizontes, which explains the periodic fevers in human malaria [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF][START_REF] Kwiatkowski | Periodic and chaotic host-parasite interactions in human malaria[END_REF]. According to [START_REF] Igor | Link between immune response and parasite synchronization in malaria[END_REF], synchronization in Plasmodium falciparum is only partial, i.e: parasiteamia often shows chaotic fluctuations on top of periodic behavior. This is confirmed by the malariatherapy patients considered in this paper. This can be seen on Fig. 6 for the patient G54: on the interval of (0,10), there are four main peaks of total parasiteamia. A high synchrony would suggest five peaks. The same remark is valid for the patients S1204 on Fig. 1 and S1050 on Fig. 7. For the patient G221, a closer look of the parasiteamia on the interval (0,10) on Fig. 9 shows also four peaks on that interval of time.

Another way to see the partial synchrony in the malariatherapy patients is to estimate the level of synchrony with the entropy formula [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF]

s(t) = 5 i=1 p i log p i
where p i is the proportion of stage i in the total infected population at time t: p i (t) = y i (t)

5 j=1 y j (t)
.

This formula suggests that if s(t) is close to zero, the population is highly synchronous (all the parasites are in the same age-class) and less synchronous otherwise (see, for instance, [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF]). The following figures represent the level of synchrony in two of our four malariatherapy patients. ) and ratio of sequestered to circulating parasites (dashed line). In this case the maximum synchrony is -1.129 while the minimum is -1.571.

Synchrony Ratio

These observation of low synchrony would serve as an additional validation of the model since it is classically observed that Plasmodium falciparum malaria is less synchronous than other form of human malaria [START_REF] Kwiatkowski | Periodic and chaotic host-parasite interactions in human malaria[END_REF][START_REF] Igor | Link between immune response and parasite synchronization in malaria[END_REF]. Also, since synchrony and the mean age of parasites define how peripheral parasiteamia may reflect total parasiteamia [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF], one can estimate the ratio of sequestered to circulating parasites, over time, as predicted by the model. When the patient is not cleared by parasiteamia, this ratio (Fig. 3) fluctuates over time. For this particular patient, after the onset of the infection, the ration is greater than 1, i.e: there are more sequestered than circulating parasites. Another interesting feature that can be seen on Fig. 3 is that the ratio increases or decreases along with synchrony after the tenth day of infection. However at the onset of the infection the level of synchrony and ratio is more disparate (ratio less than 1 while the synchrony hits two main peaks). A possible explanation of this phenomenon is that at the beginning of the infection, the correlation between sequestered and circulating parasites depends more on the mean age of parasites than on synchrony [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF].

Numerical estimation of β

Following Section 3 and by implementing the equation ( 10) on Maple, we obtain numerical estimated values of β for different patients. These estimates are given in Table 2.

# Patient β (×10 In the next section we provide an estimate using an aggregate model and compare with other values of β from the literature.

Aggregate model

The exact number of classes in the circulating or sequestered phases of parasites is not completely known. Hence, we divide the dynamics of parasite into two main classes, viz circulating parasites and sequestered parasites. Gravenor et al. [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF] have considered this decomposition as well, but they consider only the dynamics of the infected cells. Figure 5 illustrates this decomposition. Therefore, we aggregate the different infected erythrocyte stages in two stages y 1 and y 2 describing the circulating parasites that are measured and the sequestered parasites respectively. We then obtain the following aggregated model: In our control theory framework, the system (11) could be written as:

           ẋ = Λ -µ x x -βxm ẏ1 = βxm -(µ 1 + γ 1 )y 1 ẏ2 = γ 1 y 1 -(µ 2 + γ 2 )y 2 ṁ = rγ 2 y 5 -µ m m -βxm (11)
ż = A z + E d + Λ e 1 (12a) Y = C z (12b)
where:

z =       x y 1 y 2 m       , A =       -µ x 0 0 0 0 -γ 1 -µ 1 0 0 0 γ 1 -γ 2 -µ 2 0 0 0 rγ 2 -µ m       , E =       -1 1 0 -1      
and C = (0, 1, 0, 0).

The same observer ( 7) is valid. The choice of parameters is slightly easier in this case. If we consider patients that are not taking drugs, the mortality of parasite is very low [START_REF] Gravenor | An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum[END_REF]. Hence, we can consider, λ 1 = 1.03, λ 2 = 0.74 and µ 1 = 0.42, µ 2 = 0.08 . The unit remains days -1 . We choose these parameters on the basis of the precedent application. The same constraint 2 i=1

1 λ i = 1.9101065 days is satisfied. The other parameters are standards and are the same as for the above application. With the choice L = (0, 5, 0, 0) T , we have successively:

Ā = (I d -E C) A =       -µ x -γ 1 -µ 1 0 0 0 0 0 0 0 γ 1 -γ 2 -µ 2 0 0 -γ 1 -µ 1 rγ 2 -µ m       , ( Ā -LC)E =       -µ x -γ 1 -µ 1 0 0 0 -5 0 0 0 γ 1 -γ 2 -µ 2 0 0 -γ 1 -µ 1 rγ 2 -µ m             -1 1 0 -1       =       µ x -γ 1 -µ 1 -5 γ 1 -γ 1 -µ 1 + µ m       , and
L + ( Ā -LC)E =       µ x -γ 1 -µ 1 0 γ 1 -γ 1 -µ 1 + µ m      
Hence, for this model, the observer is given by:

ẇ =       -µ x -γ 1 -µ 1 0 0 0 -5 0 0 0 γ 1 -γ 2 -µ 2 0 0 -γ 1 -µ 1 rγ 2 -µ m       w +       µ x -γ 1 -µ 1 0 γ 1 -γ 1 -µ 1 + µ m       Y + Λe 1 (13a) ẑ = w + Ey (13b)
With this observer (which is independent of β), we can estimate all vector states and then the total parasite load y 1 +y 2 . Using this observer [START_REF] Kwiatkowski | Periodic and chaotic host-parasite interactions in human malaria[END_REF] with data corresponding to the four patients we obtain estimates of the total parasite load that are almost the same as those obtained with the first observer [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF]. Following [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF], we have implemented the observer using the data and then used numerical integration together with linear regression to estimate β. The obtained estimations of β for different patients with this method are given in Table 3. It can be noticed that the values (Table 3) obtained with the aggregated model are comparable to those (Table 2) obtained with the original system.

# Patient β (×10 It is worthwhile to compare these results with some others in the literature. Hetzel and Anderson [START_REF] Hetzel | The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies[END_REF] found, for rodents, β between 1.872×10 -6 (cells/µl) -1 day -1 and 7.329×10 -6 (cells/µl) -1 day -1 . Gravenor et al. in [START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF] have considered the model (1) and with a quasi steady-state assumption and by using clinical data, have estimated β = 3.4833 × 10 -6 (cells/µl) -1 day -1 . However as stated by the authors, those assumptions are simplifying. By exploiting clinical data of parasitaemia used in [START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF] in our method, we find β = 6.2734 × 10 -6 (cells/µl) -1 day -1 . According to [START_REF] Gravenor | The regulation of malaria parasitaemia: parameter estimates for a population model[END_REF], in order for the model to predict standard clinical parasitaemia, β would have to be less than 2 × 10 -6 (cells/µl) -1 day -1 (In fact β/µ m < 10 -3 in their units). Our results are consistent with their claim.

Conclusion

The estimation of sequestered parasite population has been a challenge for the biologist and modeler, with many authors [START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF][START_REF] M B Gravenor | Estimating sequestered parasite population dynamics in cerebral malaria[END_REF][START_REF] Ochola | Estimating sequestered parasite load in severe malaria patients using both host and parasite markers[END_REF] having studied this problem.

In this paper, we propose a different and simple method to estimate the total parasite concentration from the measured circulating parasites and we show how the parameter β can be estimated. To this end, we use some tools from control theory. More precisely, we built an observer, i.e: an auxiliary dynamical system whose states tend exponentially to the states of the original system (2). Moreover, this auxiliary system does not depend on the uncertain parameter β.

To summarize, a method to estimate the total parasitaemia within a malaria patient and the infection rate β has been given. This method is quite simple to implement and to use with many types of data set. Stating which of those different values of β is the best estimation is a difficult task. However our method is easy to handle and can be used for any set of data.

In conclusion, one of our main objectives here is also to show how techniques from control theory are useful to estimate parameters of epidemic and intra-host models, and are therefore of wide applicability.

Figure 1 :Figure 2 :

 12 Figure 1: Patient S1204: Estimates of the total burden with different initial conditions for the estimator (8): estimates become accurate for t > 5 days.

Figure 3 :

 3 Figure3: Patient G54: Degree of population synchrony (solid line) and ratio of sequestered to circulating parasites (dashed line). In this case the maximum synchrony is -1.129 while the minimum is -1.571.

4 Figure 4 :

 44 Figure 4: Degree of population synchrony in patient G221. The different jumps of synchrony are caused by periods of free-parasitemia in patient during the infection.

Figure 5 :

 5 Figure 5: Decomposition of Plasmodium falciparum cycle into two classes.

Figure 7 :Figure 8 :

 78 Figure 7: Patient S1050: Measure (data) of peripheral parasitaemia (solid line), and estimated total parasitaemia (dashed).

Table 1 :

 1 Estimates delivered by the estimator[START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF]

	Peripheral		Estimated sequestered	Estimated total
	parasitaemia/µl (data)	parasitaemia/µl	parasitaemia/µl
	5	5811	22727	28538
	10	180	9419	9599
	15	0	3	3
	20	9	3	12
	25	0	1	1
	30	0	1	1
	35	340	248	588
	40	25696	50217	75913
	45	336	2359	2695
	50	7660	4931	12591
	55	12900	10809	23709
	60	7100	9792	16892
	65	1420	5123	6543
	70	180	841	1021
	75	7830	6547	14377
	80	20	104	124
	85	580	2173	2753
	90	560	322	882
	95	3130	5232	8362
	100	950	1571	2521

Table 2 :

 2 Estimated values for β based on the estimator[START_REF] Mike B Gravenor | A model for estimating total parasite load in falciparum malaria patients[END_REF]

		-6 )
	S1204	0.44905
	S1050	0.72557
	G221	2.29484
	G54	1.18315

Table 3 :

 3 Estimated values for β based on the estimator[START_REF] Kwiatkowski | Periodic and chaotic host-parasite interactions in human malaria[END_REF].

		-6 )
	S1204	0.49172
	S1050	0.69842
	G221	2.36547
	G54	1.10660

Acknowledgements

We would like to thank professors M. Eichner and G. Sallet for providing us the malariatherapy data. We would also like to thank K. E. Yong for his comments. We are also very grateful to Jair Koiller and to an anonymous referee for their careful reading, and constructive comments that helped us to improve the paper.

parasitaemia per micro-liter t (days)