Fully-coupled electro-magneto-elastic behavior of unidirectional multiphased composites via finite-volume homogenization

2018

). Although a few analytical solutions are generated for electromagnetic composites under remote loading conditions [START_REF] Kuo | Fibrous composites of piezoelectric and piezomagnetic phases: generalized plane strain with transverse electromagnetic fields[END_REF][START_REF] Rashidinejad | On the exact nature of the coupled-fields of magnetoelectro-elastic ellipsoidal inclusions with non-uniform eigenfields and general anisotropy[END_REF], more advanced computational techniques require the application of more complex boundary conditions.

With the continuous development of computational capabilities and increasing demand for multiscale analysis between structures and materials, there is reviving interest in developing advanced micromechanics models to simulate the sophisticated piezoelectric-piezomagnetic coupling effects on the effective and localized responses of composite materials. For instance, [START_REF] Lee | Effective properties of three-phase electromagneto-elastic composites[END_REF] employed the finite element (FE) technique to study the averaged properties of three-phase piezoelectric-piezomagnetic composites. [START_REF] Chen | Computationally-efficient homogenization and localization of unidirectional piezoelectric composites with partially cracked interface[END_REF] studied the effect of a fully/partially-cracked interface on the stress and electric field re-distributions in unidirectional composites. [START_REF] Bishay | Computational Piezo-Grains (CPGs) for a highly-efficient micromechanical modeling of heterogeneous piezoelectric-piezomagnetic composites[END_REF][START_REF] Bishay | Multi-physics computational grains (MPCGs) for direct numerical simulation (DNS) of piezoelectric composite/porous materials and structures[END_REF] implemented the Trefftz concept into finite elements and developed the multi-physics computational grains for the analysis of both the effective and localized responses of electro-magneto-elastic composites. [START_REF] Sladek | Evaluation of effective material properties in magneto-electro-elastic composite materials[END_REF] used the meshless local Petrov-Galerkin (MLPG) technique to generate the effective properties of composites with multifunctional capabilities. Besides the numerical techniques, asymptotic homogenization methods (AHM) were also employed to study the coupled multiphysics behavior of smart heterogeneous materials [START_REF] Espinosa-Almeyda | Influence of imperfect interface and fiber distribution on the antiplane effective magneto-electro-elastic properties for fiber reinforced composites[END_REF][START_REF] Tang | Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method[END_REF]. What's more, [START_REF] Haghgoo | Fully coupled thermo-magneto-electro-elastic properties of unidirectional smart composites with a piezoelectric interphase[END_REF] employed the simplified unit cell method to investigate the effect of piezoelectric interphases on the effective multiphysics composites. [START_REF] Rabczuk | A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem[END_REF] developed a nonlocal operator method based on the variational principle in solving the dynamic eigenvalue problem of electromagnetic solids. The advantage of the nonlocal operator method is directly employing the energy principle and avoiding the mesh entanglement. Each of the aforementioned techniques has unique advantages in simulating the electro-magneto-elastic materials in terms of either the computational efforts or physical capabilities. However, most of the those techniques place more emphasis on the effective coefficients instead of the localized field concentrations under various loading/boundary conditions that should be equally important in predicting possible crack initiations which usually start from the material level.

In this contribution, we employ the finite-volume based micromechanics with multiphysics capabilities to investigate the effective and localized response of composites reinforced with electro-magnetoelastic phases. Initially employed for the fluid mechanics, the finite volume technique was introduced to solid mechanics for analysis of the micromechanical behavior of periodic composites by Pindera and coworkers [START_REF] Gattu | Parametric finite-volume micromechanics of uniaxial continuously-reinforced periodic materials with elastic phases[END_REF][START_REF] Khatam | Parametric finite-volume micromechanics of periodic materials with elastoplastic phases[END_REF], and functionally graded materials [START_REF] Cavalcante | Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials-Part I: Analysis[END_REF]. In contrast with the FE-based technique wherein the minimization of global potential energy in the unit cell with sufficient mesh refinement leads to the ultimate satisfaction of the unit cell's global equilibrium, the finite-volume technique is based on the direct satisfaction of the governing differential equations for every subvolume at each level of mesh refinement [START_REF] Chen | Homogenization and localization of elastic-plastic nanoporous materials with Gurtin-Murdoch interfaces: an assessment of computational approaches[END_REF]Chen et al., 2018bChen et al., , 2018d)). The finite-volume theory has been proved to be an attractive alternative to the finiteelement technique in micromechanical modeling of piezoelectric composites or magnetostrictive porous materials (Chen et al., 2018a;Chen and Wang, 2018). However, a fully coupled electro-magneto-elastic constitutive relationship has never been implemented into the FVDAM theory.

Starting from the microstructural characterization, repeating unit cells (RUCs) are identified with periodic boundaries. A fully anisotropic electro-magneto-elastic coupled constitutive relation is implemented into each subvolume of the multiphysics FVDAM theory. The multiphysics local stiffness matrices are constructed by relating the surfaceaveraged conjugated mechanical displacement-tractions and electric/ magnetic potential-electric/magnetic flux components. The accuracy of the theory in generating effective and localized responses are validated against the exact solutions under far-field loading conditions and the finite element simulations. More importantly, the efficiency of the corresponding program relative to the finite-element technique is tested through comparing the mesh discretization, degrees of freedom in the global system of equations, execution time, as well as the sparsities of the global stiffness matrix, where MFVDAM is superior in several perspectives. Finally, the MFVDAM is encapsulated into the particle swarm optimization algorithm to deduce optimal fiber volume fractions where the maximum electro-magneto coupling coefficients may be obtained. The identification procedure is made more precise by circumventing the traditional trial-and-error approach, substantially reducing the computational cost.

The remainder of the paper is organized as follows: Section 2 briefly introduces the theoretical framework of MFVDAM, a 0 th -order homogenization theory of fully coupled multiphysics for unidirectional composites. Section 3 derives the generalized Eshelby solutions for piezoelectric-piezomagnetic composites under multiphysics remote loading. The validation and convergence study of the proposed technique is conducted in Section 4, indicating that the MFVDAM is efficient in reducing the mesh discretization and promoting computational efficiency. Finally, the incorporation of the MFVDAM into the particle swarm optimization is illustrated in Section 5. Section 6 concludes this contribution.

Multiphysics finite-volume direct averaging micromechanics

In this contribution, the generalized fully coupled electro-magnetoelastic relation is implemented into the FVDAM theory to study the coupling effect of the fibrous composites reinforced with piezoelectricpiezomagnetic constituents. This relation can be described by the generalized Hooke's law [START_REF] Li | Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior[END_REF]:

= = + + = + + C e E q H D e E H B q E µ H ij ijkl kl kij k kij k i ikl kl ij j ij j i ikl kl ij j ij j (1)
where ij , D i and B i are the stress, electric displacement, and magnetic induction components, respectively, while ij , E i and H i are the strain components, electric and magnetic field components, respectively. C ijkl , e kij , q kij , as well as ij , µ ij and ij represent the elastic tensor, piezo- electric tensor, piezomagnetic tensor, as well as dielectric, magnetic permeability and electromagnetic tensors, respectively.

Equation ( 1) is further expressed in matrix and vector notations for an easier implementation into the FVDAM algorithm:

= D B Z E H (2)
where = Z C e q e q µ T T T .

Unit cell discretization

The multiphysics FVDAM is a simplified version of the mathematical homogenization theory or the so-called asymptotic homogenization. The latter is based on a systematic asymptotic analysis of periodic media whose response is characterized by governing differential equations with periodically varying coefficients that reflect the spatial variation of the microstructures. The mathematical homogenization theory provides a consistent framework for taking into account the effect of macroscopic strain, electric field, and magnetic field variations in a periodic material whose microstructural scale is characterized by the small parameter . Within this framework, the macroscopic and local variations of displacements, electric potential, and magnetic potential are described in a two-scale representation involving the global coordinates x for macroscopically homogeneous composite materials and the local coordinates y for the repeating unit cell, respectively, connected through the small parameter = x y / which characterizes the size of the unit cell. The operator x / i in the governing differential equations is then replaced by + x x y 1 which separates the governing differential equations into different orders of or scales. Work by Ammen et al. [START_REF] Ameen | A quantitative assessment of the scale separation limits of classical and higher-order asymptotic homogenization[END_REF] and [START_REF] He | Locally exact asymptotic homogenization of periodic materials under anti-plane shear loading[END_REF] provdes insight into the asymptotic homogenization.

The current multiphysics FVDAM formulation is developed following the classical homogenization theory or the so-called 0 th order homogenization theory. Two hypotheses have been made: the microstructural scale is infinitely small relative to the structural dimension. As a result, the strain/electric field/magnetic field gradient is neglected in the local stress/electric displacement/magnetic induction field recovery at lower microstructural scales [START_REF] Yang | Determination of metamaterial parameters by means of a homogenization approach based on asymptotic analysis[END_REF][START_REF] Zhu | Numerical investigation of necking in perforated sheets using the periodic homogenization approach[END_REF]. The mechanical displacement, electric potential, and magnetic potential in each phase of the microstructure are partitioned into average and fluctuating contributions dependent on the global and local coordinates, = x

x x x ( , , )

1 2 3 and = y y y y ( , , ) 1 2 3 , respectively, without considering the higher-order terms:

= + = + = + u x u E x H x
x y x y

x y x y

x y x y ( , ) ¯( , )

( , ) ¯( , ) ( , ) ¯( , ) i q ij j i q q j j q q j j q

( ) ( ) ( ) ( ) ( ) ( ) (3)
which produce local strain, electric field and magnetic field in the form:

= + = + + = + = = + = E E E E H H H H ¯¯¯ī j q ij ij q ij u y u y q i q i i q i y i q i i q i y ( ) ( ) 1 2 ( ) ( ) ( ) ( ) ( )
i j i i q i q i ( ) ( ) (4)
where , E ¯i and H ¯i are macroscopic strain, electric field, and magnetic field, respectively.

= u i ( 1,2,3) i q ( )
, q ( ) and q ( ) are displacement, electric potential, and magnetic potential fluctuations of the q-th subvolume, respectively. Using Eq. (4), the following expressions are obtained for the volume-averaged strain, electric field, and magnetic field taken over the analysis domain:

= + = + = + dV dV E dV E E dV H dV H H dV ¯¯V V ij ij V V ij V V i i V V i V V i i V V i 1 1 1 1 1 1
(5)

where V denotes the volume fraction of the repeating unit cell. As the fluctuating components are periodic functions on V , the integrals ap- pearing on the righthand side of Eq. ( 5) are zero, implying the average strain, electric field and magnetic field taken over the RUC are consistent with their definitions.

In the present work, we limit our analysis to unidirectional composites with continuous reinforcements, the response of which may be characterized by periodically repeating material microstructure with two-dimensional periodicity. The actual unit cell microstructure is discretized into quadrilateral subvolumes designated by the index q ( ), and its location is specified by the subvolume vertices y y ( , )

p q p q 2 ( , ) 3
( , ) referred to the fixed RUC coordinate system (the fiber direction is denoted as y 1 ). Following the convention by , the vertices are numbered in counterclockwise manners starting from the lower-left corner y y ( , )

q q 2 (1, ) 3
(1, ) and the faces are numbered in counterclockwise manners with the face F p defined by the endpoints y y ( , )

p q p q 2 ( , ) 3
( , )
and

+ + y y ( , ) p q p q 2 ( 1, ) 3 ( 1, ) for = p 1,2,3,4 such that + p yields p 1
when = p 4. The qth subvolume is generated by mapping the square-sided subvolume in the reference coordinates ( ) onto a quadrilateral subvolume resident in the actual microstructure, Fig. 1, using the Q4-type mapping function [START_REF] Cavalcante | Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials-Part I: Analysis[END_REF]:

= = = y N y i x y ( , ) ( , ) , 2,3 i q p p i p q ( ) 1 4 ( , ) (6)
where N ( , ) p is the shape function given by:

= = + = + + = + N N N N ( , ) (1
)( 1), ( , )

( , )

)( 1), ( , ) (7)

Generalized local stiffness matrix

The fluctuating displacements

= u i ( 1,2,3) i q ( )
, electric potential q ( ) and magnetic potential q ( ) in the qth subvolume are approximated using the following polynomial expansion in the reference coordinates

(

):

= + + + + = + + + + = + + + + u W W W W W W W W W W W W W W W (3 1) ( 3 1)
(

(3 1) (3 1) i q i q i q i q i q i q q q q q q q q q q q q q ( ) (00) ( ) due to the continuous reinforcement in x 1 direction. Considering the above polynomial expansion, the surface-averaged fluctuating mechanical displacements, electric potential, and magnetic potential on the pth face of qth quadrilateral subvolume are evaluated by (the superscript q is omitted for convenience):

( ) (10) ( ) (01)
= = ± = = ± = = ± + + + + + + u u u u ˆ( , 1)d ˆ( 1, )d ˆ( , 1)d ˆ( 1, )d ˆ( , 1)d ˆ( 1, )d i i i i (1,3) 1 2 1 1 (2,4) 1 2 1 1 (1,3) 1 2 1 1 (2,4) 1 2 1 1 (1,3) 1 2 1 1 (2,4) 1 2 1 1 (9)
Similar to Eq. ( 9), the surface-averaged interfacial tractions, normal electric displacements and normal magnetic induction components are defined by:

= = ± = = ± = = ± + + + + + + t t t t D D D D B B B B ˆ( , 1)d ˆ( 1, )d ˆ( , 1)d ˆ( 1, )d ˆ( , 1)d ˆ( 1, )d i i i i n n n n n n n n (1,3) 1 2 1 1 (2,4) 1 2 1 1 (1,3) 1 2 1 1 (2,4) 1 2 1 1 (1,3) 1 2 1 1 (2,4) 1 2 1 1 (10) where = t n i ji j , = D D n n j j and = B B n n j j . n n [ , ]
2 3 is the unit normal vector that defines the orientations of the surfaces of a given quadrilateral subvolume.

The surface-averaged interfacial tractions, electric displacements, and magnetic inductions in Eq. ( 10) may be expressed in terms of the surface-averaged mechanical displacements, electric potential, and magnetic potential in Eq. ( 9) through the generalized local stiffness matrices. Towards this end, we first express the 25 unknown coefficients W q .(..) ( ) in terms of the surface-averaged mechanical displacements, electric potential, and magnetic potential. The definition in Eq. ( 9) provides 20 relations, which relates the first-and second-order coefficients W q .(10) ( ) , W q .(01) ( ) and W q .(20) ( ) , W q .(02) ( ) to the surface-averaged mechanical displacements, electric potential, magnetic potential as well as the zeroth-order coefficients W .

q .(00) ( )
The determination of the zerothorder coefficients W q .(00) ( ) within each subvolume requires 5 additional equations, which are obtained through the application of equilibrium equations and Maxwell equation in the large within each subvolume:

= = = V x dV V D x dV V B x dV 1 0, 1 0, 1 0 V ij j V i i V i i (11)
This eventually leads to a local system of equations which is symbolically expressed as:

= + Y K U N Z X ˆˆq q q q q ( ) ( ) ( ) ( ) ( ) (12)
where

= Y p p p p ˆ[ ˆˆˆˆ] q q ( ) (1) (2) (3) (4) ( ) T , = t t t D B p ˆ[ˆˆˆˆˆ] p q n n p q ( , ) 1 2 3 ( , )T , = U u u u u ˆ[ ˆˆˆˆ] q q ( ) (1) (2) (3) (4) ( )T , = u u u u ˆ[ ˆˆˆˆˆ] p q p q ( , ) 1 2 3 ( , ) T , = E E E H H H X ¯[¯¯¯2¯2¯2¯¯¯¯¯¯¯] 11 22 33 23 13 12 1 2 3 1 2 3 T , = N n n n n [ ] q q ( ) (1) (2) (3) (4) ( )T , = n n n n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p q p q ( , ) 3 2 2 3 3 2 2 3 ( , )
.

The generalized local stiffness matrices K q ( ) are given explicitly by the subvolume geometry and material properties.

Global stiffness matrix

The unknown interfacial surface-averaged displacements, electric potential, and magnetic potential are determined by solving a global system of equations generated by first imposing traction, electric displacement, and magnetic induction continuity conditions at common faces of adjacent subvolumes, followed by direct enforcement of fluctuating mechanical displacement, electric potential, and magnetic potential continuity conditions. The resulting global system of equations can be symbolically represented by:

= X ˆ¯(13)
where denotes the generalized global stiffness matrix, the vector ˆis comprised of all unknown interfacial mechanical displacements, electric potentials, and magnetic potentials. The matrix represents the differences in the generalized local stiffness matrices of adjacent subvolumes.

Homogenization

Solution of the global system of equations, Eq. ( 13), produces the surface-averaged quantities U ˆ. Thus, the generalized Hill's electro- magneto-elastic concentration matrices A q ( ) for the qth subvolume is generated to relate the average strain, electric field and magnetic field to the macroscopic loading vectors (Chen et al., 2018a; Chen and Wang, 2018):

= X A X ¯q q ( ) ( ) (14)
where the elements of the concentration matrix are determined by applying one component of the macroscopic strains, electric or magnetic fields at a time. Subsequently, the averaged macroscopic stresses, electric displacement and magnetic induction components for each RUC with a volume of V are calculated as:

= = = = = = = = = = = = = = = V V v V V v V V v x x D D x D x D B B x B x B ¯( )d ( )d ( )d ( )d ( )d ( )d V V q N V q q q N q q V V q N V k q q N q q V V q N V q q q N q q 1 1 1 ( ) ( ) 1
( ) ( ) 1 1 1 ( ) ( ) 1
( ) ( ) 1 1 1 ( ) ( ) 1
( ) ( ) q q q q k q q k q (15)
where V q and v q ( ) are the volume and volume fraction of the qth subvolume. Implementation of the localization equations, Eq. ( 14), and the volume-averaged constitutive relation in each subvolume yields the effective constitutive relations for the multiphase electro-magnetoelastic fibrous composites:

= D B Z E H ¯¯¯( 16
)
where Z is the effective matrix whose elements are functions of the subvolume geometry, constituent properties, as well as the electromagneto-elastic concentration matrices. The structure of the matrix Z is expressed as:

= = = v Z C e q e q µ Z A q N q q q T T T 1 ( ) ( ) ( ) q (17)
After solving the boundary value problems and establishing the effective constitutive equations, the effective electro-magneto-elastic coefficients and localized field distributions can be efficiently generated with the present technique.

Electromagnetoelastic eshelby problem

In order to validate the MFVDAM theory, we develop the generalized Eshelby analytical solution for an electromagnetoelastic composite, which describes an infinite piezoelectric-piezomagnetic material that is embedded with a distinct electromagnetic fiber and subjected to transverse far-field electric loading D ¯2 (or magnetic loading B ¯2 ). Using the transformation relations between the Cartesian and cylindrical coordinates, the boundary conditions at infinity can be applied as:

= = = = = = D r D D r D r r B r B r ( ) ¯cos , ( ) ¯sin
( ) 0, ( ) 0 ( ) 0, ( ) 0 r zr z r 2 2 (18)
The solution of the Eshelby problem for a composite electro-magneto-elastic material that is composed of two different phases starts from the coupled constitutive relation in the cylindrical coordinate:

Table 1

Magnetoelectromechanical constants of constituent phases [START_REF] Chan | Simple model for piezoelectric ceramic/polymer 1-3 composites used in ultrasonic transducer applications[END_REF][START_REF] Li | Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior[END_REF].

11 /nCV -1 m -1 0 0 0 22 /nCV -1 m -1 0 0 0 µ 11 /nCV -1 m -1 -157 -10 0 µ 22 /nCV -1 m -1 590 -5 0
The stress equilibrium equations and the Maxwell conservation equations read:

+ + = + + = + + = 0 0 0 r r r D r r D D r B r r B B r 1 1 1 zr z zr r r r r (20)
The strain-displacement and electric/magnetic field-electric/magnetic potential relations can be expressed as: 21) into Eq. ( 19) and then into Eq. ( 20) leads to the generalized equilibrium equation and Maxwell equations in terms of the mechanical displacement, electric and magnetic potentials: By solving Eq. ( 22), the solutions for the axial displacement, transverse electric and magnetic potentials are obtained as series expansions:

= = = = = = E E H H , , , , , , zr u
= + + + = + + + = + + + u H r H r H r H r F r F r F r F r G r G r G r G r ( / )cos ( / )sin ( / )cos ( / )sin ( / )cos ( / )sin z i i i i i i i i i i i i i i i 1 3 2 4 1 3 2 4 1 3 2 4 (23)
for the fiber (i = f) and matrix (i = m) phases, respectively. It should be pointed out that the higher-order terms are eliminated by comparing the internal functions in Eq. ( 23) and far-field boundary conditions in Eq. ( 18) within which only the first-order harmonic terms (cos and sin ) are involved. In addition, we set

= = = = = = H H F F G G 0 f f f f f f 3 4 3 4 3 4
of Eq. ( 23) since the displacement, electric and magnetic potentials are bounded in the fiber domain.

By employing Eqs. ( 19) and ( 21), the corresponding axial shear stress, inplane electric displacement, and magnetic induction are expressed as:

= + = + C H e F q G C H e F q G r C H e F q G C H e F q G r D e H F G e H F G r e H F G e H F G r B q H F µ G q H F µ G r q H F µ G q H F µ G r [( ) ( ) ]cos [( ) ( ) ]sin [( ) ( ) ]cos [( ) ( ) ]sin [( ) ( ) ]cos [( ) ( ) ]sin zr i i i i i i i i i i i i i i i i i i i i i i i i i r i i i i i i i i i i i i i i i i i i i i i i i i i r i i i i i i i i i i i i i i i i i i i i i i i i i
Again, a comparison between Eq. ( 24) at infinity (r ) and the boundary conditions in Eq. ( 18) leads to (25)

+ + = = = = = = C H e F q G e H F G D q H F µ G H F G 0, ¯, 0
The remaining unknown coefficients are solved through applying the continuities at the fiber/matrix interface:

= = = = = = = = = = = = = = = = = = u r a u r a r a r a r a r a D r a D r a r a r a B r a B r a ( ) ( ), ( ) ( ) ( ) ( ), ( ) ( ( ) ( ), ( )
z f z m zr f zr m f m r f r m f m r f r m (26)
which leads to the following final system of equations:

= a a a C e q
C a e a q a e e a a a q µ q a a µ a

H F G H F G C e q e q µ H F G 1 0 0 1/ 0 0 0 1 0 0 1/ 0 0 0 1 0 0 1/ / / / / / / / / / 1 0 0 0 1 0 0 0 1 f f f m m m f f f m m m f f f m m m f f
The unknown coefficients can be determined from Eqs. ( 25) and ( 27). The solution is employed to validate the present MFVDAM technique in the next section.

Numerical results

As explained in Sections 2-3, the MFVDAM enforces volume-averaged equilibrium and conservation equations within each subvolume and adopts surface-averaged continuity conditions between adjacent elements (Fig. 1), while the finite-element methods are based on the variation of total Lagrange multipliers and node-to-node continuities. Hence, the equilibrium equations and conservation equations are not satisfied at the element level in the FE simulation. Validating the FVDAM-generated results against the Multiphysics FEM and Eshelby solutions in both the effective response and localized elastic-electricmagnetic concentrations provides robust credence of the developed MFVDAM results. In the meantime, the computational efficiency and convergence of the present approach relative to the finite-element results are illustrated.

The constituents of the fibrous composites studied in this section are composed of the CoFe 2 O 4 matrix reinforced with piezoceramic BaTiO 3 fibers. The material constants found in the reference [START_REF] Li | Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior[END_REF] are listed in Table 1. It should be noted that although the materials investigated herein are transversely isotropic, the MFVDAM admits fully anisotropic electro-magneto-elastic phases.

Analytical verification

The multiphysics FVDAM are firstly validated by comparison with the exact solution for a dilute case where an electro-magneto-elastic fiber is embedded in an infinite matrix which is further subjected to farfield loading conditions. This is the generalized Eshelby problem described in Section 3. To simulate the far-field loading in the analytical solution, a square repeating unit cell with 5% fiber volume fraction is adopted in the MFVDAM simulation. The detailed mesh discretization is illustrated in Fig. 2 with a zoom-in screen of the meshes.

In this situation, a far-field unit electric displacement is applied in the transverse direction

= D 1 C/m 2 2
, where the resultant distributions of electric displacement D y y ( , ) 2 2 3 , magnetic induction B y y ( , ) 3 2 3 , as well as the axial shear stress component y y ( , ) 12 2 3 are compared between the derived Eshelby solutions and MFVDAM predictions. As observed in Fig. 3, the differences between the two computational approaches are almost invisible. We note that while the magnitude of electric displacement D y y ( , ) 2 2 3 in the fibrous domain is much higher than that in the matrix phase, almost zero values are observed for the magnetic and stress fields in the fiber domain. In a more direct comparison, the converted radial electric displacement D y y ( , ) r 2 3 and magnetic induction B y y ( , ) r 2 3 , as well as the stress y y ( , ) zr 2 3 from the Cartesian coordinates are quantitatively compared at the fiber/matrix interface, Fig. 4. The MFVDAM simulation coincides with the exact solution almost everywhere of the entire angular position. A close look at the axial shear stress reveals a slight difference and the stress magnitudes predicted by the Eshelby solutions are slightly larger than the MFVDAM results. The differences between the elasticity and numerical solutions can be attributed to the applied boundary conditions. While far-field separable loading condition is applied in the former case (Section 3), the periodic boundary conditions are implemented in the latter one (Section 2). We have verified that the difference vanishes when a smaller fiber volume fraction is utilized.

Numerical verification

After ensuring the accuracy of the localized stress/electric/magnetic field distributions, the MFVDAM-generated effective coefficients are tested against the in-house finite-element results [START_REF] Chen | Computationally-efficient homogenization and localization of unidirectional piezoelectric composites with partially cracked interface[END_REF], which provide a gold standard. It is important to point out that the finite-element results have been generated in an uncompiled MA-TLAB environment that mimics the unit cell solution methodology employed by the multiphysics FVDAM theory, where the mechanical Therefore, the two sets of results may be compared on a one-to-one basis. We note that there are two fundamental differences between the finite-element and multiphysics FVDAM approaches in the manner of satisfying the governing differential equations and applying the continuity and periodicity conditions. Firstly, while the minimization of total potential energy within the finite-element framework leads to the ultimate satisfaction of the unit cell's global equilibrium and conservation with sufficient mesh refinement, the MFVDAM approach enforces equilibrium and conservation in the integral sense for every subvolume at each level of mesh refinement. Secondly, the continuity and periodicity conditions are enforced only on the adjacent nodal displacements and electric/magnetic potentials of the discretized unit cells in the finite-element approach, whereas the continuity and periodicity conditions are applied on both the surface-averaged tractions, electric displacement, magnetic induction and mechanical displacements, electric/magnetic potentials in the multiphysics finite-volume technique.

First of all, selected homogenized elastic moduli were generated for a BaTiO 3 /CoFe 2 O 4 material system with both square and hexagonal fiber configurations for fiber volume fractions in the range of [0,0.65] f , Fig. 5. Perfectly matched results are obtained between the finite-element and finite-volume techniques, providing solid credence of the present technique. The selected moduli vary linearly with increasing fiber content, including the transverse normal and shear moduli.

Variations of the effective piezoelectric and piezomagnetic coefficients as a function of fiber content are illustrated in Fig. 6. The piezoelectric/piezomagnetic coefficients are used to measure the conversion between the mechanical energy and the electric/magnetic energy and they are thus important indicators for the energy-converting effect. Accurate prediction of the corresponding coefficients helps to understand the composites' application in the smart structures. Since the BaTiO 3 fiber is a piezoelectric material while the CoFe 2 O 4 is a piezomagnetic material, the effective piezoelectric coefficients increase from zeros with the increase of the fiber volume fraction, while the effective piezomagnetic coefficients exhibit an opposite trend. More interesting results are generated in Fig. 7 for the dielectric, permeability, and electromagnetic coefficients. It is reported in the literature that the coupling electromagnetic coefficients ij , which are important to indicate the energy conversion between electricity and magnetism, are difficult to obtain for any monoclinic material. However, the present results suggest that the non-zero electromagnetic coefficients can be generated in composite materials even though the corresponding coefficients are zeros in either constituent, Table 1. Once again, the finite-volume and finite-element methods generate highly agreeable results.

It is important to note that the microstructural effect is negligible on either mechanical or electric/magneto properties, especially at low and intermediate volume fractions. This is due to the fact that, for the present material system, the property contrast between the BaTiO 3 and CoFe 2 O 4 phases is insignificant. The array effect becomes noticeable at larger fiber-volume fractions where the fiber-fiber interaction is more pronounced. The multiphysics FVDAM is hence consistent with this observation, demonstrating the method is sufficiently sensitive to capture these small effects.

We proceed to test the convergence of the multiphysics FVDAM for generating the local stress/magnetic/electric field distributions as a function of mesh discretization. Three different mesh discretizations are employed with progressively greater refinement. Fig. 8 illustrates the meshes of a hexagonal repeating unit cell with 180, 720, and 2880 subvolumes, respectively, all of which are implemented into the inhouse FE programs to generate reference results. The fiber volume fraction is fixed at 0.25 in this simulation. The corresponding degrees of freedom (DOFs) of the three meshes in the two computational approaches are listed in Table 2, where MFVDAM usually consumes twothirds of the total DOFs by MFEM. Fig. 9 illustrates the non-zero element distribution within the global stiffness matrix for the mesh C discretization. Due to the applied periodic boundary conditions, the global stiffness matrix is sparse but not banded. Therefore, inverting the global stiffness becomes more computationally demanding. The number of non-zero elements in the global system of equations of MFVDAM approach is drastically smaller than that of the MFEM of the same mesh discretization, with a concomitant decrease of density of the nonzero elements in the global system of equations in the case of MFVDAM calculation, hence an improvement in computational efficiency. The execution time by the MFEM and MFVDAM based on three runs are enclosed in Table 2 for higher-order internal trial functions employed in the finite-element scheme, the convergence of the local fields generated by the MFEM with mesh refinement is faster than that of MFVDAM technique. The MFVDAM technique requires a greater mesh discretization to achieve the same smoothness of the local stress/electric/magnetic fields as those of the MFEM, where the discontinuity of stress/field gradients becomes invisible only for the finest mesh discretization. Nonetheless, the basic characters of the local stress/magnetic/electric fields generated by the MFVDAM are captured with sufficient accuracy even in the case of a coarse mesh (Mesh A), providing additional evidence of the proposed multiphysics FVDAM technique.

In the application of multiphysics composites, the interfacial bonding state plays an important role in affecting the localized responses and hence the homogenized properties of the composites. Based on the credence of the previous simulation, we further investigate the effect of interfacial microcracks on the electrical displacement D 3 distribution of a square unit cell subjected to unit magnetic loading = B ¯1 N/Am 2 , Fig. 12. The fiber volume fraction is prescribed as 20%. A pair of symmetric cracks were prescribed along the ± °30 arc segment of the interface. The continuities of displacements and electric/magnetic potentials are no longer satisfied at the cracked interface. It is observed that the microcracks along the interface significantly affects the internal electric-displacement distributions. Meanwhile, a singular electric field appears at the crack tips, leading to further damage development. The FVDAM results coincide with the FEM prediction, even at the crack tips.

Experimental verification

= + V (1 ) f f m f , where = f m ( , )
i is the density of either the fiber or matrix phase, Table 1. c) The longitudinal acoustic velocity is related to material piezoelectricity and density: = C ¯/ , and S ji is the effective compliance compo- nent and calculated from

= S C [ ] [ ] 1 .
Fig. 13 compares all of the above four calculated parameters against the experimental data (Chan and Unsworth, 1989) for PZT-7A/Araldite D composites. The elastic and piezoelectric properties of PZT-7A fiber are listed in Table 1 with the density of the PZT-7A prescribed as 7700 kg/m 3 , while the properties of the isotropic Araldite D matrix are: Young's modulus = E 4.877GPa, Poisson's ratio = v 0.355 , and density = 1150kg/m 3 . It should be noted that the measured value d 11 for a randomly chosen PZT-7A ceramic is 163-167 × 10 -12 m/V. Hence, f but with a diminishing rate. In the meantime, the acoustic impedance keeps increasing almost a constant speed. The observed phenomenon also offers a general idea of designing the multiphysics composites through tailoring the microstructural details (such as fiber volume fraction and constituent properties) to achieve the optimal parameters, which is introduced in the next section.

Application

The interaction between magnetism, electricity and elasticity in the unidirectional piezoelectric-piezomagnetic composites produces a unique property that reflects the magnetoelectric coupling This suggests potential usage in magneto-electro-mechanical transducers and sensors. It has been already recognized that the fiber volume fraction has a remarkable influence on the magnetoelectric coupling coefficients of the composites. However, the change of these magnetoelectric coupling coefficients in terms of the fiber volume fraction cannot be described using a simple formula. Herein, the identification of the accurate fiber volume fraction that yields the maximum magnetoelectric coupling coefficients is accomplished by incorporating the MFVDAM theory into the particle swarm optimization algorithm in order to avoid tedious computations in the traditional trial-and-error approach. The MFVDAM-driven particle swarm optimization algorithm facilitates the rapid search of the candidate solutions in all possible fiber volume fraction space via minimizing the objective function: because the value of 22 and 33 are negligible when compared with the value of a . The magnified a can thus be obtained by minimizing the objective function in Eq. ( 28).

The particle swarm optimization is advantageous in the sense that it avoids the gradient matrix inversion that could cause numerical instability and instead adopts the concept of social behavior of birds/ants in the process of food search. The general idea is to initiate a group of swarm population whose positions are the initial candidate solutions that will be optimized. For instance, the i-th particle of the population is denoted by its position x m and moving velocity v m , respectively. The initial position of the i-th particle carries the information of the guessed fiber volume fraction. The updating algorithm of the i-th particle at k +1-th iteration is thus expressed as:

= + + + x x v m k m k m k 1 1 (29)
where the moving velocity consists of three components, including its velocity at k-th iteration, the i-th particle's local best experience and the population's global best experience:

= + + + v wv c p x c g x [ rand()( ) rand()( )] m k m k m k m k k m k 1 1 2 (30)
where p m k and g m k are the local global best experiences, respectively. c 1 and c 1 are the acceleration numbers, respectively. rand() is a random number between [0,[START_REF] Rashidinejad | On the exact nature of the coupled-fields of magnetoelectro-elastic ellipsoidal inclusions with non-uniform eigenfields and general anisotropy[END_REF]. w is weight factor to constrain the velocity of the last step and is defined in terms of its initial and final values. is introduced in Eq. ( 30) to ensure convergence by tailoring the updating distance.

A group of 15 particles is created within a range between [0,0.75], each particle represents the potential value of the parameter to be optimized -fiber volume fraction. A maximum of 20 iterations is set as the stopping criterion to prevent the ceaseless loop. The detailed updating algorithm is referred to Fig. 14. In this particular situation, we set the error tolerance as 0.01, which is an extremely demanding case, and thus the optimization procedure is continuously updated until the maximum iteration number is reached. A final optimized fiber volume fraction of = V 0.4429

f optimum
is obtained as the magnetoelectric coupling coefficient is magnified to 2.789 CV -1 m -1 from zeros from both composite phases. Fig. 15 tests the optimization's computational efficiency by generating the objective function as a function of iteration numbers and illustrating the population's locations after several different iteration steps. It is seen that the optimization is almost converged to a value of OBJ = 0.3586 after the fourth iteration, which is also proved by Fig. 15(b) where a few particles already reach the lowest point after the 5-th iteration and all particles converge to the same point after the final iteration.

Summary and conclusions

The 0-th order finite volume direct averaging micromechanics is extended with fully coupled piezoelectric-pizeomagnetic capabilities to simulate the effective and localized responses of periodic unidirectional composites. For composites with generalized constituents, the present theory admits fully anisotropic material constitutive relationship for subvolumes. The repeating unit cells (RUCs) with square or hexagonal geometry are adopted to mimic different fiber arrangements. The present MFVDAM theory is validated against the in-house higher-order finite elements and generalized Eshelby technique. The computational efficiency of the MFVDAM is tested by directly comparing against the in-house MFEM in the effort of implementing mesh discretization, stiffness matrix densities as well as the execution time. It is concluded that the present MFVDAM can significantly enhance the computational effort without sacrificing accuracy. We also conclude that the electromagnetic coupling coefficients can be generated by combining pure piezoelectric and piezomagnetic constituents, which offers more flexible options in the design of electric-magnetic conversion micromachines.

Fig. 1 .

 1 Fig. 1. Parametric mapping of a reference square subvolume in the coordinate system onto a quadrilateral subvolume in the y y 2 3 coordinate of the actual microstructure.

Fig. 2 .

 2 Fig. 2. A repeating unit cell containing 0.05 fiber content in square array and a detailed close-up used for comparison with the generalized Eshelby solution for an inclusion embedded in an infinite matrix.

Fig. 3 . 1

 31 Fig. 3. Comparison of selected local field distribution in the region occupied by the unit cell obtained from the generalized Eshelby and dilute multiphysics FVDAM solutions during far-field loading = D 1 C/m 2

Fig. 4 .

 4 Fig. 4. Comparison of radial electric displacement D r , magnetic flux density B r , and shear stress zr , around the fiber/matrix interface obtained from the generalized Eshelby and dilute multiphysics FVDAM solutions.

Fig. 5 .

 5 Fig. 5. Selected homogenized elastic moduli of a unidirectional BaTiO 3 /CoFe 2 O 4 composite as a function of BaTiO3 volume content.

Fig. 6 .

 6 Fig. 6. The effective piezoelectric and piezomagnetic coefficients of a unidirectional BaTiO 3 /CoFe 2 O 4 composite as a function of BaTiO3 volume content.

Fig. 7 .

 7 Fig. 7. The dielectric, permeability and electromagnetic coefficients of a unidirectional BaTiO 3 /CoFe 2 O 4 composite as a function of BaTiO3 volume content.

Fig. 8 .

 8 Fig. 8. Discretizations of hexagonal unit cells with different mesh refinements.

Fig. 9 .

 9 Fig. 9. Nonzero element distributions in the global stiffness matrix for the type-C mesh.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Comparison of transverse magnetic flux densityB 2 distribution with imposition of unit macroscopic. = B ¯1 N/Am 2

  Fig. 12. Effect of interfacial bonding state on the electric displacement D 3 distribution with imposition of unit macroscopic = B ¯1 N/Am 2 generated by the MFVDAM (left) and MFEM (right).

Figures

  Figures of merits are used to assess the utility of piezoelectric composites in specific application scenarios. Various parameters are defined to evaluate the performance of the piezoelectric composites (Chen et al., 2018a): a) The piezoelectric coupling constant: = C C 1 / t 11 11 , where = + C C e ¯/ 11 11 11 11 . b) The acoustic impedance = Z C , where is the density of the composite given by = + V (1 )

Fig. 13 .

 13 Fig. 13. Comparison of figure of merits generated by the in-house multiphysics FVDAM theory and experimental data (Chan and Unsworth, 1989).

  are the weights of the three magnetoelectric coupling coeffi- cients. Herein, we set =

Fig. 14 .

 14 Fig. 14. Updating algorithm of the MFVDAM-based optimization.

Fig. 15 .

 15 Fig. 15. Illustration of computational efficiency of the optimization: (a) objective function as a function of iteration numbers and (b) convergence of population after four iteration steps.

Table 2

 2 Comparison of the convergence between the present MFVDAM and finite element technique.

	Meshes	Discretization	Degrees of freedom		Nonzero elements		Density		Execution time a /s
			MFVDAM	MFEM	MFVDAM	MFEM	MFVDAM	MFEM	MFVDAM	MFEM
	A	180	1800	2700	32376	76430	0.9993%	1.0484%	1.86	4.65
	B	720	7200	10800	129271	305174	0.2494%	0.2616%	5.11	53.29
	C	2880	28800	43200	516815	1219535	0.0623%	0.0653%	162	846

a Averaged based on three runs.

Acknowledgement

The corresponding author is supported by the National Natural Science Foundation of China (No. 12002303) and Fundamental Research Funds for the Central Universities (2020QNA4016).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Vincent Henry

Ivan Moszer

Olivier Dameron

Laura Vila Xicota

Bruno Dubois

Marie-Claude Potier

Martin Hofmann-Apitius

Olivier Colliot
email: olivier.colliot@sorbonne-universite.fr

Converting disease maps into heavyweight ontologies: general methodology and application to Alzheimer's disease

Keywords: Ontology, Systems Medicine Disease Map, Knowledge model, Data integration, Alzheimer's disease

Omics technologies offer great promises for improving our understanding of diseases. The integration and interpretation of such data pose major challenges, calling for adequate knowledge models. Disease maps provide curated knowledge about disorders' pathophysiology at the molecular level adapted to omics measurements. However, the expressiveness of disease maps could be increased to help avoiding ambiguities and misinterpretations and to reinforce their interoperability with other knowledge resources.

Ontologies are an adequate framework to overcome this limitation, through their axiomatic definitions and logical reasoning properties. We introduce the Disease Map Ontology (DMO), an ontological upper model based on systems biology terms. We then propose to apply DMO to Alzheimer's disease (AD). Specifically, we use it to drive the conversion of AlzPathway, a disease map devoted to Alzheimer's disease, into a formal ontology: AD Map Ontology (ADMO). We demonstrate that it allows one to deal with issues related to redundancy, naming, consistency, process classification and pathway relationships. Furthermore, we show that it can store and manage multi-omics data. Finally, we expand the model using elements from other resources, such as clinical features contained in the ADO (AD Ontology), resulting in an enriched model called ADMO-plus. The current versions of DMO, ADMO and ADMOplus are freely available at http://bioportal.bioontology.org/ontologies/ADMO.

Introduction

Systems medicine disease maps (DM) provide curated and integrated knowledge on pathophysiology of disorders at the molecular and phenotypic levels [START_REF] Mazein | Systems medicine disease maps: community-driven comprehensive representation of disease mechanisms[END_REF]. Based on a systems biology approach, they describe all biological physical entities (i.e. gene, mRNA, protein, metabolite) in their different states (e.g. phosphorylated protein, molecular complex, degraded molecule) and the interactions between them. Their relations are represented as molecular interactions (as well as covalent modifications) organized in pathways, which encode the transitions between participants' states as processes [START_REF] Mazein | Systems medicine disease maps: community-driven comprehensive representation of disease mechanisms[END_REF] [START_REF] Fabregat | The Reactome Pathway Knowledgebase[END_REF]. Most advanced DM projects focus on Parkinson's disease [START_REF] Fujita | Integrating pathways of Parkinson's disease in a molecular interaction map[END_REF], cancer [START_REF] Kondratova | Signalling maps in cancer research: construction and data analysis[END_REF], rheumatoid arthritis [START_REF] Singh | Computational Systems Biology Approach for the Study of Rheumatoid Arthritis: From a Molecular Map to a Dynamical Model[END_REF] [START_REF] Singh | RA-map: building a state-of-the-art interactive knowledge base for rheumatoid arthritis[END_REF], asthma [START_REF] Mazein | AsthmaMap: An expert-driven computational representation of disease mechanisms[END_REF], atherosclerosis [START_REF] Parton | New models of atherosclerosis and multi-drug therapeutic interventions[END_REF], macrophage activation transduction signaling [START_REF] Raza | Construction of a large scale integrated map of macrophage pathogen recognition and effector systems[END_REF] and Alzheimer's disease (AD) [START_REF] Ogishima | AlzPathway, an Updated Map of Curated Signaling Pathways: Towards Deciphering Alzheimer's Disease Pathogenesis[END_REF].

AD is a progressive neurodegenerative disorder of the brain, which was first described in 1906. The intense activity of AD research constantly generates new data and knowledge on AD-specific molecular and cellular processes (a Medline search for "Alzheimer's disease" results in over 115,000 articles, as of December 2019). However, the complexity of AD pathophysiology is still imperfectly understood [START_REF]2019 Alzheimer's disease facts and figures[END_REF]. These 110 years of efforts have essentially resulted in one dominant paradigm to underline the causes of AD: the amyloid cascade [START_REF] Golde | Anti-aβ therapeutics in Alzheimer's disease: the need for a paradigm shift[END_REF]. Nevertheless, therapeutics targeting this pathway failed to lead to curative outcome for humans, leading to the need for additional hypotheses [START_REF] Herrup | The case for rejecting the amyloid cascade hypothesis[END_REF]. Briefly, several approaches have been pursued in order to target the amyloid metabolic cascade for treatment of AD [START_REF] Briggs | Drug treatments in Alzheimer's disease[END_REF]. Among them, there have been treatments targeting BACE-1 that proved to lower Aβ production and brain amyloid load in animal models [START_REF] Eketjäll | AZD3293: A Novel, Orally Active BACE1 Inhibitor with High Potency and Permeability and Markedly Slow Off-Rate Kinetics[END_REF], but did not show any improvement in cognition in clinical trials [START_REF] Wessels | Efficacy and Safety of Lanabecestat for Treatment of Early and Mild Alzheimer Disease: The AMARANTH and DAYBREAK-ALZ Randomized Clinical Trials[END_REF] (lanabecestat), or even worsened symptoms [START_REF] Egan | Randomized Trial of Verubecestat for Mild-to-Moderate Alzheimer's Disease[END_REF]. Similar results were found for drugs that targeted the -secretase, for immunization approaches, or for treatment with monoclonal antibodies. This could be explained by the fact that Aβ accumulation is a gradual process that takes many years to occur, and is linked to changes in the macro-and microenvironment of the brain and the neurons, including neuroinflammation, alterations in endolysosomal trafficking, tau accumulation, membrane cholesterol changes. Therefore, stopping the amyloid cascade when the environment is already altered might not be sufficient to improve cognitive deficits or to stop of the cognitive decline. In conclusion, treatments should potentially start before the apparition of cognitive signs and should likely be combined with treatments targeting other mechanisms.

Since the turn of the century, omics technologies (genomics, transcriptomics, proteomics, phosphorylomics, metabolomics…) lead to a more comprehensive characterization of biological systems and diseases. The production of omics data in AD research thereby opens promising perspectives to identify alternatives to the amyloid cascade paradigm. There is a clear need to integrate the amyloid cascade as a component of the whole organ-wide dysregulation occurring in AD, rather than treating it as an isolated component. Therefore, a model should be built that integrates tau, neuroinflammation, cholesterol metabolism, insulin resistance, neuronal degeneration, and all the other known pathways involved in Alzheimer's disease. The current challenge is to connect and integrate these data in an appropriate way.

AlzPathway is a DM developed for AD [START_REF] Ogishima | AlzPathway, an Updated Map of Curated Signaling Pathways: Towards Deciphering Alzheimer's Disease Pathogenesis[END_REF]. Although very rich in AD-specific pathophysiology information (it describes 1,347 biological physical entities, 129 phenotypes, 1,070 biochemical reactions and 26 pathways), this resource does not provide sufficient formalism to adequately interlink current knowledge and omics data: it would thus benefit from a refined level of description, able to cope with the complex modeling of disease systems and the diversity of measurements from biomedical experiments. This lack of formalism is inherent to all disease maps. The information contained in DM is stored in syntactic formats developed for systems biology: the Systems Biology Graphical Notation (SBGN) [START_REF] Rougny | Systems Biology Graphical Notation: Process Description language Level 1 Version 2.0[END_REF], the modified Edinburg Pathway Notation (mEPN) [START_REF] Freeman | The mEPN scheme: an intuitive and flexible graphical system for rendering biological pathways[END_REF] and the Systems Biology Markup Language (SBML) [START_REF] Smith | SBML Level 3 package: Hierarchical Model Composition, Version 1 Release 3[END_REF].

While syntactic formats are able to index information and can be managed by different applications such as MINERVA [START_REF] Gawron | MINERVA-a platform for visualization and curation of molecular interaction networks[END_REF] or NaviCell [START_REF] Kuperstein | NaviCell: a web-based environment for navigation, curation and maintenance of large molecular interaction maps[END_REF], they are not expressive enough to define explicit relationships and formal descriptions, leading to possible errors and misinterpretations (e.g. reaction "re1178" is describing the translation of the IL1B gene into IL1B mRNA; this description does not allow to interpret whether re1178 is a transcription instead of a translation or whether IL1B gene is a transcript instead of a gene). For AlzPathway, this defect in expressiveness results in a lack of: a) hierarchy and disjunction between species (e.g. between "Protein" and "phosphorylated Protein" or between "Protein" and "RNA", respectively), b) formal definition of entities (such as phenotypes), c) formal relationships between reactions and pathways, d) uniformity of entities' naming (e.g. complexes that are labelled by their molecular components or by a common name) and e) consistency between reactions and their participants (e.g. translation of genes instead of transcripts).

Compared to syntactic formats, the Web Ontology Language (OWL), a semantic format used in ontologies, has higher expressiveness [START_REF] Schaffert | A semantic wiki for collaborative knowledge formation[END_REF] and was designed to support knowledge and data integration. Moreover, OWL combines high expressivity and logical constraints to ensure the consistency of the resource [START_REF] Noy | Representing the NCI Thesaurus in OWL DL: Modeling tools help modeling languages[END_REF]. It is thus a good candidate to overcome the previous limitations. An ontology is an explicit specification of a set of concepts and their relationships, represented in a knowledge graph in semantic format.

Ontologies provide a formal naming and definition of the types (i.e. the classes) and interrelationships between entities (i.e. the properties) that exist for a particular domain.

Moreover, knowledge and data managed by an ontology benefit from their logical semantics and axiomatic properties (e.g. class disjunction, cardinality, existentiality, universality), which supports automatic control of consistency and additional information inferences (including hierarchy and relationships) [START_REF] Mizoguchi | Part 1: Introduction to ontological engineering[END_REF].

In the biomolecular domain, the Gene Ontology (GO) provides the community with the largest set of controlled vocabulary to index and share data [START_REF]The Gene Ontology Resource: 20 years and still GOing strong[END_REF]. WikiPathways [START_REF] Slenter | WikiPathways: a multifaceted pathway database bridging metabolomics to other omics research[END_REF] and the Systems Biology Ontology (SBO [START_REF] Courtot | Controlled vocabularies and semantics in systems biology[END_REF]) also provide controlled vocabulary hierarchies.

But none of these ontologies provide enough specificity for AD pathophysiology. In the AD domain, the Alzheimer's Disease Ontology (ADO) [START_REF] Malhotra | ADO: a disease ontology representing the domain knowledge specific to Alzheimer's disease[END_REF] organizes information describing clinical, experimental and molecular features in OWL format for text mining. However, the description of the molecular systems of ADO is less specific than that of AlzPathway.

In this paper, we propose the Disease Map Ontology (DMO), an ontological upper model able to drive the conversion of a disease map into a formal ontology. We then apply it to convert AlzPathway into an OWL ontology which we call the Alzheimer Disease Map Ontology (ADMO). Finally, we show that ADMO can be connected with ADO into ADMOplus, a resource able to store and interconnect biomedical data. These different steps are summarized in Figure 1.

Ontological upper model: Disease Map Ontology

The first task (Figure 1A) aimed at designing a generic ontological upper model able to drive the conversion of the specific content of a disease map (in our case AlzPathway). In an expressive ontology, the relationships are not only links between classes, but also logical constraints (i.e. axioms) that are inherited by all their descendants (subclasses). Thus, the choices of axioms that support high level classes and their properties are key elements for the usefulness of the model.

Design of DMO classes

SBO [START_REF] Courtot | Controlled vocabularies and semantics in systems biology[END_REF] is a terminology that provides a set of classes commonly used to index information in SBML format. These classes conceptualize biological entities with a suitable balance between genericity and specificity in order to improve the genericity of representation despite the diversity of reactions: thus systemic models can be adequately represented using few classes, while preserving a satisfactory level of accuracy, similarly to the BioModels representation [START_REF] Malik-Sheriff | BioModels-15 years of sharing computational models in life science[END_REF]. To build the DMO ontological model, we first selected SBO terms from "process" or "material entity" classes that fit with DM content, specifically those corresponding to the reaction types present in the map legend. This resulted in 54 terms: 37 reaction types and 17 molecule types, respectively. Then, we relied on the vocabulary used to define components' shapes in the graphical format mEPN, to complete the SBO class set with molecular states that fit with DM knowledge (e.g. phosphorylated or truncated). Following class selection from SBO and mEPN, we designed a class hierarchy between them. Classes related to participants were separated in two hierarchies: one describing their biochemical properties such as polypeptide chains, simple chemicals, genes or non-covalent complexes, one describing the state of participants such as native form, phosphorylated or truncated. We systematically added disjointness constraints between the generic sibling subclasses of participants in the biochemical property hierarchy in order to ensure that process participants belong to only one set (e.g. a gene cannot be a protein and reciprocally). We did not apply the same rule to the state hierarchy as, for instance, a truncated protein could also be phosphorylated. Classes related to processes were also hierarchized without disjointness constraints as a reaction may refer to different processes (e.g. a transfer is an addition and a removal).

Design of DMO properties

Properties consistent with a systems approach (i.e. has_part, has_component, has_component_process, has_participant, has_input, has_output, has_active_participant and their respective inverse properties) were selected from the upper-level Relation Ontology (RO) [START_REF] Smith | Relations in Biomedical Ontologies[END_REF]. Then, we enriched the formal definition of our set of process classes with these properties and associated cardinalities to link processes and participants with relationships in description logic (e.g. a transcription has at least one gene as input and at least one mRNA as output; a protein complex formation has at least two proteins as input and at least one protein complex as output).

Finally, four other properties were added: occurs_in (a property selected from RO) to link a process to its respective location, derives_from to link a modified protein to its initial form, has_template (sub-property of derives_from) to link a mRNA to its related gene or a protein to its related mRNA, and has_mutation (sub-property of has_part) to link a gene to its possible mutations.

DMO design results

The design of the DMO upper ontological model based on SBO, mEPN, RO and de novo additions resulted in 143 classes (43 processes' subclasses and 83 participants' subclasses) and 14 properties formally defined by 188 logical axioms in description logic (Figure 2). This model is based on a simple pattern as our knowledge graph involves only four types of properties (and their inverse properties): 1) the is_a (subclass_of) standard property, 2) the has_part standard property and its sub-properties has_component, has_component_process and has_mutation 3) the has_participant property and its subproperties has_input, has_output and has_active_participant and 4) the location property occurs_in.

In AlzPathway, native and modified proteins (e.g. phosphorylated or activated) have the same name and differ only in their graphical shapes. In order to specify these different states, we added a suffix to modified protein labels (e.g. "_P" or "_a" for phosphorylated or activated, respectively).

In AlzPathway, phenotypes are participants. But several of them are named with a process name, pathway label or molecule type (e.g. microglial activation, apoptosis or cytokines, respectively). In order to deal with these ambiguities, 26 phenotypes were reclassified as molecules (e.g. cytokine) or cellular components (e.g. membrane) and 14 names that referred to processes or pathways were changed into processes' participant names (e.g. 'apoptosis' that refers to a process was changed into 'apoptotic signal' that refers to a participant). In addition, 5 phenotypes that were named with a pathway name (e.g. apoptosis)

were added to the initial set of the 26 AlzPathway's pathways. AlzPathway describes a subset of genes, mRNAs and proteins, but not always the whole combination of one given gene, its related mRNAs and proteins. As omics technology can capture data at the genome, transcriptome or proteome levels, we added missing genes, mRNAs or proteins in order to always have the description of the gene, the mRNA and the protein for a same entity described in AlzPathway. Additional entities were linked with the has_template relationships but not linked to reactions when no corresponding knowledge was found in AlzPathway. In such a way, we avoided overinterpretations. This resulted in the addition of 407 genes, 416 mRNAs and 191 proteins.

AlzPathway conversion in OWL format

Then, using the Protégé ontology editor (a free, open-source, software that provides a convenient interface to edit OWL files, widely used in research and supported by an international community) [START_REF] Musen | The protégé project: a look back and a look forward[END_REF], the content of the structured table was imported into DMO and converted in OWL using the Protégé Cellfie plugin. AlzPathway's molecular and phenotypic entities were integrated as subclasses of DMO "participant" classes without redundancies (172 redundant participants were identified). Reactions extracted from AlzPathway were integrated as independent subclasses of the "process" class, without hierarchy. Then, automatic reasoning was used to classify them as subclasses of the DMO upper model process classes depending on their formal definition (see Figure 3a*), independently of their initial types in AlzPathway. The 1,065 inferred SubClassOf axioms were added to the ontology.

In the original paper, AlzPathway is described as a resource containing reactions and their corresponding pathways. Nevertheless, in AlzPathway, the reactions are not formally linked to corresponding pathways because the pathways are described as free text. We created classes corresponding to pathways by transforming the free text information into formal classes. Thus, we manually created classes corresponding to these pathways. Then, they were automatically linked to relevant reactions using description logic: for each pathway class, a class "reaction involved in pathway x" was created and defined both as a "reaction that has_participant the molecules of interest in x" and as a "component_process_of pathway x".

For example, the class "reaction involved in WNT signaling pathway" has_participant "WNT" and is a component_process_of "WNT signaling pathway". Then, using automatic reasoning, all reactions having participants involved in pathway x were classified as subclasses of the "component_process_of pathway x" classes and were linked to the pathway by subsumption with the component_process_of property. For example, "SFRP-WNT association" is automatically classified as subclass of "reaction involved in WNT signaling pathway" (see Figure 3b*) and inherits from its property: component_ process_of "WNT signaling pathway" (see Figure 3b**). The 355 inferred SubClassOf axioms corresponding to reactions involved in one of the 22 pathways were added to the ontology. This resulted in an extended version of DMO, specific to AD physiopathology: ADMO. Finally, in order to catch Single Nucleotide Polymorphisms (SNPs) measurements, we also added 7,523 classes corresponding to SNPs from the NeuroChip SNP microarray [START_REF] Blauwendraat | NeuroChip, an updated version of the NeuroX genotyping platform to rapidly screen for variants associated with neurological diseases[END_REF], which are related to genes described in AlzPathway.

Results: ADMO content

Building ADMO resulted in a consistent network containing 2,132 classes (1,175 disjoint participants, including 88 phenotypes or signals, 1,038 reactions and 22 pathways) linked with 10,964 logical axioms before and 12,373 logical axioms after automatic reasoning. The conversion of AlzPathway benefited from the DMO simple pattern of relationships (Figure 4A). Specific efforts were dedicated to the formal definition of the network with description logic axioms, leading to explicit relationships between processes, biological entities and pathways. These axiomatic definitions resulted in an increase of formalism compared to the initial representation of AlzPathway information. Following automatic reasoning, only 15 out of 643 AlzPathway's reactions generically considered as 'transition' or 'unknown transition' remained unassigned to a specific process of the DMO upper model (e.g. 'metabolic reaction', 'phosphorylation' or 'activation'). Moreover, 41 processes in AlzPathway were consistently assigned to a specific process different from their initial consideration (such as translation instead of transcription) and were, therefore, manually corrected. In addition, 355 reactions were formally defined as subprocesses of pathways thanks to automatic reasoning.

Connection with ADO and use for storage of biomedical data:

ADMO-plus

Mapping of ADMO with ADO

ADMO is a formal representation of AD pathophysiology at the molecular scale. It was designed to store and link omics biomedical data. Nevertheless, it would be interesting to also link data from other scales such as brain imaging or clinical scores (Figure 1C). ADO [START_REF] Malhotra | ADO: a disease ontology representing the domain knowledge specific to Alzheimer's disease[END_REF] describes knowledge not only about molecular processes (as in AlzPathway) but also about clinical assessments. By converting and integrating AlzPathway in OWL format, the resulting ontology and ADO are represented in the same format, and thus can be connected with each other. In the first step, we selected ADO classes that correspond to ADMO ones. ADO classes were imported into ADMO independently of their initial hierarchy. Then, they were defined either a) as equivalent classes of ADMO "process", "pathway", "phenotype" or "gene" classes (e.g. ADO: "Abeta-RAGE interaction" class is equivalent to ADMO: "AB-RAGE_complexation" class) or b) with DMO relationships towards ADMO classes (e.g. ADO: "macrophage activation" class is equivalent to has_output ADMO: "activated microglia" class or ADO: "neuron process" class is equivalent to occurs_in ADMO: "neuron" or "neuron compartment" classes). Thus, for equivalent classes, ADO imported classes inherited from ADMO definition (e.g. ADO: "Abeta RAGE interaction" class inherits the ADMO "AB-RAGE_complexation" class definition: 'protein-protein complexation' that has for input ADMO: 'RAGE' and ADMO: 'Amyloid decamere' and has for output ADMO: 'AB-RAGE'). For newly defined classes, automatic reasoning made it possible to build a new hierarchy between ADO and ADMO classes. All in all, 32 ADO classes were imported into ADMO (suppl. Table 1-ADO) resulting in ADMO-plus.

Biomedical data integration

Ontologies' classes can be filled by representative individual instances (a task called "instantiation"), which allows them to be used as resources for data storage. Thus, the next step consisted in instantiating ADMO-plus with biomedical omics data. As a proof of principle, SNP, gene expression (transcriptomic) and metabolomic data from the INSIGHT-PreAD study were used as instances to fill ADMO-plus classes (Figure 4B). The INSIGHT-PreAD is an ongoing prospective monocentric cohort with the objective to determine factors that increase the risk of progression of cognitively normal old adults to clinical AD. The study was approved by the local ethical committee (ANSM 130134B-31) and all participants signed a written informed consent. More information on the study is available in Supplementary text 1. As a proof of concept, we selected INSIGHT-PreAD genotypic, transcriptomic and metabolomic data that presented significant score variation. Among these data, only 16 SNPs corresponding to 11 genes (out of 53 SNPs corresponding to 44 genes), 23 mRNA relative expression (out of 145) and 25 metabolomic data (out of 53) could be integrated as classes' instances in ADMO-plus. They were typed by their corresponding classes (for instance SNP, RNA or metabolite, see Figure 4B), and inherited from classes' properties and thus from reaction and pathway information contained in the ADMO-plus network.

Discussion

We proposed the DMO ontological upper model in order to drive the conversion and integration of disease maps into formal ontologies. We demonstrated its utility by converting AlzPathway [START_REF] Ogishima | AlzPathway, an Updated Map of Curated Signaling Pathways: Towards Deciphering Alzheimer's Disease Pathogenesis[END_REF] into an ontological model, called ADMO. It provides an increase in formalism, makes it interoperable with other ontologies (such as ADO [START_REF] Malhotra | ADO: a disease ontology representing the domain knowledge specific to Alzheimer's disease[END_REF], GO [START_REF]The Gene Ontology Resource: 20 years and still GOing strong[END_REF], the Protein Ontology [START_REF] Natale | Protein Ontology (PRO): enhancing and scaling up the representation of protein entities[END_REF]) and makes it able to integrate biomedical data. Based on a systems biology paradigm [START_REF] Kitano | Foundations of systems biology[END_REF], all ADMO entities are formally defined as classes and interconnected within a consistent network. While AlzPathway contained several ambiguities, our efforts on formalism using description logic in the definition of ADMO classes allowed us to solve inconsistencies and provide a precise specification of processes and biological entities within the system. To our knowledge there is no previous work on DM conversion to OWL and this idea of converting an existing resource into an ontology is new.

Formalization of the fine description provided by DM.

The increased formalism requires to assert a participant as a subclass of the most representative class and thus clarifies the status of the entities. In several standard bioinformatics knowledge resources (e.g. UnitProt [START_REF]UniProt: the universal protein knowledgebase[END_REF], KEGG [START_REF] Kanehisa | KEGG: new perspectives on genomes, pathways, diseases and drugs[END_REF]), a same ID refers to a gene or a protein and in fine to a set of information, such as interactions, regulations and posttranslation modifications (PTM), which are thus not specifically discriminated. However, omics technologies are able to generate data focused on specific elements of the systems (gene mutation, relative gene expression, protein concentration, ubiquitination ratio, phosophorylation ratio, etc.). When compared to other graph resources [START_REF] Briggs | Drug treatments in Alzheimer's disease[END_REF] [START_REF] Rougny | Systems Biology Graphical Notation: Process Description language Level 1 Version 2.0[END_REF][41] that focus on genes and reactions only, DM take each part of the system into consideration from genes expression to PTM [START_REF] Mazein | Systems medicine disease maps: community-driven comprehensive representation of disease mechanisms[END_REF]. Our ADMO proposition goes one step further by formally defining the different elements of the system. By providing disjoint classes for different molecular states, DMO breaks ambiguities between genes, genes product and their modified states [START_REF] Slenter | WikiPathways: a multifaceted pathway database bridging metabolomics to other omics research[END_REF].

ADMO can be instantiated with omics data within the specific corresponding classes, resulting in an ontology that explicitly integrates each type of omics data despite the complexity of the AD pathophysiology system.

Automatic reasoning facilitated by the systems paradigm

Taking advantage of a systems biology approach and reasoning properties, DMO can automatically ensure satisfiability of ADMO, and provides inferences of hierarchy and new relationships (such as the link between a pathway and its process components) [START_REF] Mizoguchi | Part 1: Introduction to ontological engineering[END_REF]. Other ontologies provide generic models in the field of molecular biology, such as BioPax [START_REF] Demir | The BioPAX community standard for pathway data sharing[END_REF].

BioPax is a well-established framework to share information between knowledge bases. However, it was essentially designed to manage knowledge sharing, and logical reasoning is limited to satisfiability check. While DMO presents a level of genericity similar to that of BioPax, it is designed to manage subclasses (e.g. ADMO) that in turn manage data as their instances. Moreover, due to its management by automatic logical reasoning and its model based on systems paradigm, ADMO is particularly flexible. If reaction, participant or pathway classes are removed or added, automatic reasoning is able to rebuild the modified network in a consistent way. In addition, class instantiation facilitates this task and instanced data take advantage of an up-to-date network. In existing resources, biological entities are mainly annotated by the cellular component, molecular function or biological process classes from GO [START_REF] Balakrishnan | A guide to best practices for Gene Ontology (GO) manual annotation[END_REF]. While this provides the largest set of data indexed by a controlled vocabulary [START_REF]The Gene Ontology Resource: 20 years and still GOing strong[END_REF], this is also limited by the fact that genes are not gene products, nor functions or processes.

Thus, genes cannot be assigned as individual instances of GO classes, leading to an underuse of the reasoning ability provided by ontologies. With the classical annotation methods, a knowledge change, such as the addition or the removal of a new molecular reaction, involves a verification of all annotated entities that were implicitly related to the process concerned.

With instantiation methods, relationships between participants and processes are explicitly expressed as axioms and a modification in the ontology directly applies to instanced entities.

Finally, automatic reasoning also provides formal relationships between reactions and their related pathways, which did not exist in AlzPathway. The formal specification of pathways and their relationships with reactions in ADMO opens new promises for mapping ADMO with widely-used ontologies such as GO or the Human Phenotype Ontology (HPO) [START_REF] Köhler | The Human Phenotype Ontology in 2017[END_REF].

Connection with other resources and potential extensions

Converted in OWL format, AlzPathway's knowledge is now interoperable with other ontologies such as ADO [START_REF] Malhotra | ADO: a disease ontology representing the domain knowledge specific to Alzheimer's disease[END_REF]. This results in a mutual enrichment: ADMO is linked with clinical knowledge and ADO benefits from more specific knowledge about AD pathophysiology. Here, we present a first attempt to connect ADMO/ADO into ADMO-plus, as a proof of concept. Going further would necessitate a revision of ADO, which is designed for text mining and which is not adapted to logical reasoning and systems biology. The OWL format also opens perspectives to integrate other knowledge resources. Here, we relied on AlzPathway, but additional resources could be used. In the domain of AD, the knowledge graph neuroRDF [START_REF] Iyappan | NeuroRDF: semantic integration of highly curated data to prioritize biomarker candidates in Alzheimer's disease[END_REF] would increase current knowledge provided by AlzPathway and ADO.

Our DMO upper ontological model also provides an interesting framework to embed generic resources and thus harmonize AlzPathway and those resources. This offers new avenues for increasing the scale of representation of AD pathophysiology network in our framework. Indeed, our approach shall facilitate the future update of AlzPathway. Ontologies manage knowledge in a network as a directed graph. Thus, when reactions are added or removed, they are automatically integrated within or suppressed from the network. As ADMO is based on automatic reasoning for classification, any new reaction described with its reactants will automatically be described hierarchically by its type and linked to a previously described pathway. Nevertheless, the integration of biomedical data from the INSIGHT-PreAD study underlined the fact that the range of data identified as significant in this study is larger than the coverage of ADMO-plus (which reflects the current knowledge specific to AD pathophysiology). While AlzPathway currently provides the larger network in the domain, it is a compilation of knowledge from a reductionism approach. This suggests that the network has to be extended in order to generate new hypotheses about AD etiology [START_REF] Herrup | The case for rejecting the amyloid cascade hypothesis[END_REF].

Our strategy could be applied to other DM and increase their interoperability. The main limitation is the availability of DM of interests. Apart from that, we believe that this work can be applied to other DM since DMO was designed to embed any DM and can easily be reused. In practice, there will be specific parts that will need to be adapted, specifically the extraction of the DM content and its possible modifications made by a domain expert. Then, other tasks can be easily done using the Protégé editor.

Even though they are less specific than DM, considering generic systems resources, such as Reactome [START_REF] Fabregat | The Reactome Pathway Knowledgebase[END_REF], would provide useful additional information through a wide range of generic curated biochemical reactions and pathways. The integration, in the same framework, of disease-specific and non-specific pathways is useful to build a more comprehensive view of the disease, in particular if these pathways are interconnected. When such integration is done, OWL format allows to tag pathways as disease-specific or not, so that the user can be clear about their status, leading to an enrichment without adding noise. In the same way, the genericity of processes and participants described in the DMO upper model opens the perspective to harmonize specific DM with an equivalent curation level from other neurodegenerative disorders, such as the Parkinson's disease map [START_REF] Fujita | Integrating pathways of Parkinson's disease in a molecular interaction map[END_REF].

Differences with DM

Unlike DM [START_REF] Mazein | Systems medicine disease maps: community-driven comprehensive representation of disease mechanisms[END_REF], ontologies are not adapted to graphical visualization [START_REF] Kitano | Using process diagrams for the graphical representation of biological networks[END_REF]. Thus, DM are better adapted to human reading. On the other hand, ontologies can store the network (classes and relationships) in a more consistent way and can thus be tested for logical consistency. Also, they present a higher flexibility to formally integrate new elements in the knowledge graph, as we did by adding 865 genes, related SNPs and mRNAs. Note that, during the conversion step, AlzPathway's internal IDs were retained as class annotations, allowing interoperability and retrieval between the initial (graphical) and converted (formal) resource. Thus, DM and ontologies are complementary approaches. The combination of these two approaches will be beneficial for both the DM and bio-ontology communities. Types (i.e. the classes): collection of sets that can be unambiguously defined by a property that all its members share. Universality: a universal quantification is a type of quantifier which is interpreted as "given any" or "for all". It expresses that a propositional function is satisfied by every member of a domain of discourse.

FiguresFigure 1 .

 1 Figures

  Figure 1. Summary of the workflow for AlzPathway conversion in OWL, from the DMO

Figure 2

 2 Figure 2 Disease Map Ontology (DMO) model design. Term of classes were extracted from

Figure 3 .

 3 Figure 3. Example of automatic reasoning with Protégé. Asserted axioms are shown in

Study concepts and study design: VH, IM, OC Acquisition, analysis or interpretation of data interpretation: all authors Manuscript drafting or manuscript revision for important intellectual content: all authors Approval of final version of submitted manuscript: all authors research: VH Obtained funding: OC, IM, VH Study supervision: OC, IM

ontologies. We demonstrated its use through the conversion of an Alzheimer's disease map. This enriches it with reasoning properties and makes it interoperable with other ontologies. This should constitute a useful resource for the community. The approach is generic and can be applied to other disease maps.

Source of funding

The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6), from the Inria Project Lab Program (project Neuromarkers) and from the Fondation Vaincre Alzheimer (grant number FR-18006CB).

Role of the funding source

The sponsors had no role in study design, data analysis or interpretation, writing or decision to submit the report for publication.

AlzPathway conversion driven by DMO: the Alzheimer Disease

Map Ontology

DMO was designed to integrate DM knowledge as subclasses and manage its consistency and formalism. Here, we demonstrate its use in the case of the conversion of the AlzPathway DM into a formal ontology, ADMO (Alzheimer Disease Map Ontology; Figure 1B).

Extraction of AlzPathway contents

AlzPathway information, contained in the original SBML file [START_REF] Mizuno | AlzPathway: a comprehensive map of signaling pathways of Alzheimer's disease[END_REF], was exported using CellDesigner [START_REF] Funahashi | CellDesigner 3.5: A Versatile Modeling Tool for Biochemical Networks[END_REF] in a tabular format, which was further restructured using home-made Python scripts (suppl. Fig 1). This step involved both manual and automatic transformations.

We thus created a table (suppl. Table 1) in which each biological entity was indexed by one of the DMO participants' subclasses and all processes were matched with their input, output or active participants. In AlzPathway, reactions have no naming: they are labeled from r1 to r1070. To facilitate human readability, processes were labelled with a concatenation of the type of the reactions and the names of the participants. The table was also supplemented with class annotations corresponding to other information contained in AlzPathway such as the AlzPathway identifier (ID), and IDs from other knowledge bases like UniProt [START_REF]UniProt: the universal protein knowledgebase[END_REF] for participants and KEGG [START_REF] Kanehisa | KEGG: new perspectives on genomes, pathways, diseases and drugs[END_REF] for processes. The table was structured to integrate component information in case of multiplex entities (e.g. protein complexes) or the initial (native) entity in case of modified entities (e.g. phosphorylated or truncated proteins), and location information for processes (e.g. cell type or cell part). The table was then manually curated as described below.

AlzPathway content modification and addition

Authors contributions

Dr Vincent Henry had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Existential restriction: an existential quantification is a type of quantifier which is interpreted as "there exists", "there is at least one", or "for some". Expressiveness: the expressive power of a language is the breadth of ideas that can be represented and communicated in that language. The more expressive a language is, the greater the variety and quantity of ideas it can be used to represent.

Relationships: relationships (also known as relations) between objects in an ontology specify how objects are related to other objects. Typically, a relation is of a particular type (or class) that specifies in what sense the object is related to the other object in the ontology.

Logical axioms: assertions (including rules) in a logical form that, together, comprise the overall theory that the ontology describes in its domain of application.

Supplementary Text 1

INSIGHT-PreAD enrolled participants (318) aged 70 to 85 years, with a subjective cognitive decline (SCD) and no objective cognitive disorders defined by a mini-mental state examination score (MMSE) ≥ 27 and total recall score in the free and cued selective reminding test (FCSRT) ≥ 41. Exclusion criteria included clinical dementia rating scale (CDR) > 0, visual and auditory functions insufficient for neuropsychological testing, the existence of a known neurological disease, recent stroke and illiteracy. SNP genotyping was performed with genomic DNA extracted from blood cells using the Illumina NeuroChip, a low-cost, custom-designed array containing a tagging variant backbone of about 306,670 variants complemented with a manually curated custom content comprised of 179,467 variants implicated in diverse neurological diseases, including Alzheimer's disease, Parkinson's disease, Lewy body dementia, amyotrophic lateral sclerosis, frontotemporal dementia, progressive supranuclear palsy, corticobasal degeneration, and multiple system atrophy. Transcriptomic, metabolomic and lipidomic data were obtained from 96 INSIGHT subjects as described previously (PMID31492558).

Supplementary Figure 1 Supplementary Table 1 https://zenodo.org/record/4545641#.YC0cpi17SqQ