Priscilla Keip
email: priscilla.keip@cirad.fr

Sébastien Ferré
email: ferre@irisa.fr

Alain Gutierrez

Marianne Huchard
email: marianne.huchard@lirmm.fr

Pierre Silvie
email: pierre.silvie@cirad.fr

Pierre Martin
email: pierre.martin@cirad.fr

S Litu

W Pros

V Fusc

V Parv

A Indi

C Spp

C Opul

C Papa

D Dume

Z Mays

A Escu

B Ole

G Hirs

S Lyco

R Com

× S Spp

Z Mays × × ×

S Litto

× S Lyco ×

S Litu+spp

B Oler

× S Spp+any

S Litto+spp

W Prost

Practical Comparison of FCA Extensions to Model Indeterminate Value of Ternary Data

Keywords: Graph-FCA, Relational Concept Analysis, Triadic Concept Analysis

The Knomana knowledge base brings together knowledge from the scientific literature on the use of plants with pesticidal or antibiotic effects on animals, plants, and human beings to propose protection solutions using local plants. In this literature, the elements of the 3-tuple (protected organism, protecting plant, pest) are named using the binomial nomenclature consisting of the genus name followed by the species name. In some instances, authors use the abbreviation "sp." in the singular or "spp." in the plural, as species name, to indicate the indeterminate status of the species for a guaranteed genus. To suggest protection solutions, the indeterminacy of the species has to be hypothesized based on assigning the sp./spp. to the other species in the same genus and conversely. This paper discusses the classification of ternary data containing some indeterminate values generated by three extensions of Formal Concept Analysis.

Introduction

According to the Committee on World Food Security, governments have to improve small food producers' access to knowledge on the use of biodiversity for crop protection to enhance food security and nutrition [START_REF] Hlpe | Agroecological and other innovative approaches for sustainable agriculture and food systems that enhance food security and nutrition[END_REF]. With the aim of proposing protection solutions based on the use of local plants, the Knomana Knowledge Base (KKB) brings together knowledge from the scientific literature on the use of plants with pesticidal or antibiotic effects on animals, plants, and human beings [START_REF] Martin | Knowing the Useful Plants for Organic agriculture according to literature: Building and Exploring a Knowledge Base for Plant and Animal Health[END_REF]. The methodology used to suggest protection solutions is based on Formal Concept Analysis [START_REF] Keip | Effects of Input Data Formalisation in Relational Concept Analysis for a Data Model with a Ternary Relation[END_REF].

In KKB, a plant use is characterized using 70 descriptors, for example, the part of plant used to prepare the bio-product, the method of preparation and its efficacy. In early January 2020, KKB contained 42,400 plant use descriptions from 630 documents, dated 1957 to 2019. These descriptions include a total of 1,782 plants to protect 64 (crop, animal and human) systems against 451 bio-aggressors (pests, diseases, bacteria, viruses, etc.). This 3-tuple (protected organism, protecting plant, pest 5 ) is the core of the protection solution as its efficacy is linked to the interactions between these three elements.

In the literature, each element of the 3-tuple is named using the binomial nomenclature, which consists of the genus name followed by the species name. This designation is a widely accepted convention for the formal scientific naming of the biological organisms, and avoids the ambiguity associated to vernacular names that are not shared by all communities. For the name of some species within the 3-tuples, authors use the abbreviation "sp." in the singular or "spp." in the plural. The abbreviation sp. indicates that the genus name of the organism is known by the authors but its species name is not; the abbreviation spp. indicates that at least two organisms of the same genus are considered without their names being mentioned.

To classify 3-tuples using Formal Concept Analysis (FCA), the presence of the sp. or spp. abbreviation in a formal context corresponds to the absence of the name of the species, and thus to indeterminate value of data. This indeterminacy has to be hypothesized to suggest protection solutions. This paper deals with ternary relationships with FCA and discusses how to account for the indeterminate specification of species. Three extensions of FCA, Triadic Concept Analysis (TCA) [START_REF] Lehmann | A Triadic Approach to Formal Concept Analysis[END_REF], Relational Concept Analysis (RCA) [START_REF] Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF], and Graph-FCA (GFCA) [START_REF] Ferré | Graph-FCA: An extension of formal concept analysis to knowledge graphs[END_REF], were qualitatively and quantitatively evaluated on two reduced KKB datasets.

Section 2 presents a short excerpt of KKB to show the two issues addressed in this paper. Sections 3, 4 and 5 present solutions using, respectively, TCA, RCA, and GFCA. Section 6 compares the different solutions and Section 7 concludes the paper.

Indeterminate Information and Ternary Relationships in the Knomana Knowledge Base

For this work, two datasets were extracted from KKB. They describe known protection solutions against species of Spodoptera, including the pest Spodoptera frugiperda that is currently spreading very rapidly in Africa and is devastating maize, tomato and cabbage crops. The first dataset is composed of 12 3tuples describing relations between six organisms, nine plants, and three pests (S. littoralis, S. litura, and Spodoptera spp.). This dataset enabled our qualitative evaluation that consisted in comparing the classifications provided by the FCA extensions to highlight similarities and differences among them. The second dataset was used for the quantitative evaluation of scalability. It is composed of 34 3-tuples describing relations between six organisms, 30 plants and four pests, Spodoptera spp., S. frugiperda, S. littoralis, and S. litura. This dataset includes the first one composed of three pests.

In both these datasets, the pests all belong to the genus Spodoptera. One is called Spodoptera spp., showing it refers to at least two species of the Spodoptera genus. It is not mentioned if the species are S. littoralis, S. litura, and S. frugiperda or others. To deal with this indeterminate value, we defined the three levels of aggregation presented in Fig. 1. The known species and the Spodoptera spp. indeterminate species are located at the bottom of the hierarchy. The hypothetical species, resulting from the aggregation of the ones located at the bottom, are located above: S. litu+spp (resp. S. litto+spp) gathers information on S.litu (resp. S. litto) and S.spp. That is, it encompasses known information on S. litu (resp. S. litto) or at least two Spodoptera species, possibly including S.litu (resp. S. litto). At the top of the hierarchy, S. spp+any gathers information on S. litto, S. litu, and S. spp. The use of this hierarchy differs depending on the FCA extension concerned and is described in the following sections.

Another characteristic of KKB is a central ternary relationship connecting the three main components: protected organisms, protecting plants and pests. FCA [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF] and RCA consider binary relationships that require application of conceptual model transformations as proposed in the database domain [8]. The two other FCA extensions studied, TCA and Graph-FCA, are directly applicable to this kind of data: TCA because it was introduced for this purpose, Graph-FCA because it allows the definition of n-ary relationships of any dimension.

Both datasets and all files, corresponding to each evaluation of the three next sections, are available online [START_REF] Martin | Dataset on Spodoptera used to conduct the practical comparison of FCA extensions to Model Indeterminate Value of Ternary Data[END_REF].

Triadic Concept Analysis (TCA)

Triadic concept analysis (TCA) was introduced by Lehmann and Wille to deal with situations where "an object g has the attribute m under the condition b" [START_REF] Lehmann | A Triadic Approach to Formal Concept Analysis[END_REF]. By extension, it can be considered to deal with any ternary relationship. In our case, the ternary relationship can be embedded in such a scheme. Pro-tected organisms, protecting plants, and pests can be used respectively as objects, attributes and conditions. A triadic context is a 4-tuple K = (G, M, B, Y ) where G is the set of objects, M the set of attributes, B the set of conditions, and Y ⊆ G × M × B associates an object and an attribute under a condition. For example, Table 1 (the three tables in the top row) is a triadic context in which the objects are protected organisms, the attributes are protecting plants, and the conditions are pests. It associates object A.escu with attribute A.indi under condition S.litto, meaning that A.indi treats A.escu when attacked by S.litto. Let us denote this relationship IT , and IT [P rotected,P lant] the mapping of IT on the first two components. Triplets inferred from this initial knowledge with the addition of S.litto+spp, S.litu+spp and S.spp+any, to deal with indeterminate value, are presented in the three tables in the bottom row in Table 1: ∀(po, pl, pe) ∈ IT [P rotected,P lant] × {S.litto, S.litu}, (po, pl, pe + spp); ∀(po, pl, pe) ∈ IT [P rotected,P lant] × {S.spp}, (po, pl, S.litto + spp) and (po, pl, S.litu + spp); ∀(po, pl) ∈ IT [P rotected,P lant] , (po, pl, S.spp + any). It is easy to generalize this addition to species other than S.litto and S.litu.

A triadic concept (A1, A2, A3) of (G, M, B, Y ) satisfies A1 ⊆ G, A2 ⊆ M and A3 ⊆ B as well as:

A1 × A2 × A3 ⊆ Y ((A1, A2, A3) is a rectangular parallelepiped full of ×); and X1 × X2 × X3 ⊆ Y , A1 ⊆ X1, A2 ⊆ X2 and A3 ⊆ X3 implies that (A1, A2, A3) = (X1, X2, X3) (formal concept max- imality).
A1, A2 and A3 are respectively called extent, intent and modus of the triadic concept (A1, A2, A3). Table 1 highlights in red the triadic concept TC5 = ({A.escu, B.ole}, {A.indi, C.papa}, {S.litto, S.litto + spp, S.spp + any}). Selected triadic concepts for the triadic context of Table 1 are presented in Table 2 and were built using FCA Tools Bundle6 . They reveal several facts on the Spodoptera excerpt. We chose them because they are non-trivial and provide valuable information on data. TC2 indicates that no plant protects all organisms against Spodoptera. TC5 groups A.escu and B.ole, which are protected against S.litto by A.indi and C.papa. This shows that A.indi and C.papa can replace one another to control S.litto. As shown by TC6, A. indi controls some Spodoptera species on the three plants A.escu, B.ole and Z.mays.

Relational Concept Analysis (RCA)

Relational Concept Analysis (RCA) [START_REF] Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF] was developed to view a dataset (called a relational context family, or RCF) as a set of objects of several categories (e.g. protecting plants, protected organisms and pests), connected through several relationships (e.g. an organism is protected by a plant and a plant treats a pest). Three object-attribute contexts (Plants and Protected, with nominal scaling, and Pests with hierarchical scaling) describe the three object categories, while two object-object contexts (protectedBy and treats) encode the relationships. These relationships contain the same information as that listed in Table 1. RCA acts as an iterative process with several steps. Relations are considered through the addition of relational attributes based on quantifiers inspired by description logic operators [START_REF]The Description Logic Handbook: Theory, Implementation, and Applications[END_REF]. The detailed process is described in [START_REF] Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF] and for this paper, we used the RCAExplore tool [START_REF] Dolques | Rcaexplore, a FCA based tool to explore relational data[END_REF] available online 7 .

Figure 2 shows an excerpt of the result. The concept number in the lattices is respectively 12 (protected organisms), 14 (plants), and seven (pests). Concept CP lant13 (Fig. are illustrated with quantifier ∃. The conceptual models corresponding to these variants are shown on the left in Figure 4. A first alternative (called RCAR for RCA with Reification) consists of reifying the ternary relation that is thus added as a fourth object-attribute context to the RCF, whose objects are the 3-tuples of the relation. This results in 4 lattices. Figure 3 shows an excerpt of the central lattice (24 concepts) where the objects are the 3-tuples. The other lattices are quite flat, or with a hierarchy (the case of pests). Concept CP ion22 groups the 3-tuples corresponding to Triadic Concept TC5, and the relational attribute ∃manages(CP est1) (S.litto). Their respective plants and protected organisms are separated into sub-concepts of CP ion22. Triadic concept TC6 corresponds to CP ion19, and the relational attributes indicate that S.litto+spp is managed (∃manages(CP est4)) and the plant used is A.indi (∃uses(CP lant1)). A second alternative (called RCAG for RCA with Graph view) consists of an RCF composed of two categories that describe protected organisms and pests, respectively, and one relationship per protecting plant. This results in two lattices, with nine protected organism concepts (Fig. 4 (middle)), and seven pest concepts (Fig. 4 (right)). Triadic concept TC5 (resp. TC6) appears as CP rot8 (resp. CP rot9). The shared relational attributes describe protecting plants as roles pointing to controlled pests.

Graph-FCA (GFCA)

Graph-FCA (or GFCA) [START_REF] Ferré | Graph-FCA: An extension of formal concept analysis to knowledge graphs[END_REF] was introduced as an extension of classical FCA to knowledge graphs, or to be more precise, to directed multi-hyper-graphs. A graph context is a 3-tuple K = (O, A, I) like in FCA except that objects play the role of nodes in the graph, and elements of the incidence relation I are directed hyper-edges connecting objects and labeled with attributes. Unary attributes connect a single node: they act as node labels and correspond to classical FCA attributes. Binary attributes connect one node to another, and therefore represent relationships or links between objects. N-ary attributes can encode more complex relationships involving more than two participants (e.g., "plant X protects organism Y against a pest Z"). The intents of graph concepts are like conjunctive queries, i.e. graph patterns with a distinguished node. GFCA also defines n-ary graph concepts, but here we only consider unary concepts.

V.cane(CPest5) ∃ V.cane(CPest3) ∃ V.cane(CPest2) ∃ V.cane(CPest0) ∃ V.fusca(CPest5) ∃ V.fusca(CPest3) ∃ V.fusca(CPest2) ∃ V.fusca(CPest0) ∃ V.parvi(CPest5) ∃ V.parvi(CPest3) ∃ V.parvi(CPest2) ∃ V.parvi(CPest0) ∃ C.spp(CPest2) ∃ C.spp(CPest0) ∃ C.spp(CPest1) ∃ C.opu(CPest2) ∃ C.opu(CPest0) ∃ C.opu(CPest1) ∃ W.pro(CPest3) ∃ W.pro(CPest0) ∃ W.pro(CPest4) ∃ W.pro(CPest1) CProt2 B.ole B.ole CProt3 G.hirsu ∃ D.dume(CPest4) ∃ D.dume(CPest1) ∃ D.dume(CPest6) G.hirsu CProt6 R.com ∃ W.pro(CPest5) ∃ W.pro(CPest2) ∃ W.pro(CPest6) R.com CProt4 S.lyco ∃ V.cane(CPest4) ∃ V.cane(CPest1) ∃ V.cane(CPest6) ∃ V.fusca(CPest4) ∃ V.fusca(CPest1) ∃ V.fusca(CPest6) ∃ V.parvi(CPest4) ∃ V.parvi(CPest1) ∃ V.parvi(CPest6) S.lyco
Choosing a Graph-FCA representation of the 3-tuples (Protected,Plant,Pest) amounts to choosing what the objects and the attributes are. An immediate representation (GFCA-Ternary) would be to introduce a ternary attribute for the ternary relation between protected organisms, protecting plants, and pests: e.g., protection(A.escu, A.indi, S.litto). Unary attributes are used with nominal scaling on organisms and plants, and with hierarchical scaling on pests: e.g., A.indi (A.indi), S .spp any(S.litto). As a result, we obtain 36 concepts (excluding bottom concepts): 13 concerning protected organisms, 17 concerning plants, and 6 concerning pests. They are grouped in three connected graph patterns: pattern Q1 is specific to the 3-tuple (R.com,W.pro,S.litu), and corresponds to triadic concept TC1; pattern Q3 contains all the other 3-tuples because they are all interconnected; the concepts in pattern Q2 generalize the concepts of the two other patterns.

Figure 5 shows an excerpt of GFCA output8 . For instance, concept Q2g characterizes the four organisms (out of six) that are protected by plants (Q2d) that protect against S.litu+spp (Q2c). It contains two organisms, A.escu and B.ole, that are not known to be protected against S.litu+spp by any of those plants (they are not in concept Q2a, a sub-concept of Q2g). The concerned plants, i.e. C.opu, C.spp, A.indi, W.pro, are therefore candidates for protection of the two organisms against S.litu+spp. The triadic concepts can be found in the GFCA output. TC2 corresponds to concept Q2e, which characterizes the set of protecting plants, showing that no plant protects all organisms because its intent contains no specific plant. TC5 corresponds to concept Q3u, which states that A.indi and C.papa have in common to protect both A.escu (Concept Q3a) and B.ole (Concept Q3b) against S.litto (Concept Q3n). TC6 corresponds to concept Q3w, which is the most general concept of the organisms that are specifically protected by A.indi. Its extent contains A.escu, B.ole, and Z.mays; and its intent states that the protection is against S.litto+spp.

A more compact representation of the 3-tuples (GFCA-Binary) would be to use plants as binary relations from organisms to pests: e.g. A.indi (A.escu, S.litto). There are several possible justifications for this repre-Fig. 6. Concept lattices and graph patterns for GFCA-Binary sentation. First, protecting plants is the focus of the study, and we prefer concept intents that contain explicit references to plants: using them as attributes is one way to enforce this. Second, the plants concerned can be considered as agents intervening between organisms and pests. This representation resembles that of RDF triples where the middle element is used as an edge label [START_REF] Hitzler | Foundations of Semantic Web Technologies[END_REF]. We applied the same scaling as above for organisms and pests, and simple hierarchical scaling for plants with a binary edge plant(X, Z) for every 3-tuple (X, Y, Z). Figure 6 shows the complete GFCA output, here with a graphical representa- tion of intents: solid arrows represent the edges of the graph patterns that define concept intents. We obtained 18 concepts grouped in six patterns: 12 concerning organisms, and 6 concerning pests. Protecting plants are displayed as edge labels. The exclusive use of unary and binary attributes makes the output much easier to read. Despite the simplification, triadic concepts can still be found in the GFCA-Binary output, e.g.: TC2 as the edge from Concept Q3b to Q3a; TC5 as the edge from Q6g to Q6a; TC6 as the edge from Q5b to Q5a.

Results of the Whole Use Case and Discussion

This section presents the results of the complete Spodoptera dataset, which is almost three times larger than the first one (an increase from 12 to 34 3-tuples).

We then discuss the different methods.

Results of the complete Spodoptera dataset. The second dataset contains 34 3tuples using six organisms, 30 plants, and four pests, while the first one has 12 3-tuples, using respectively, six, nine and three species. Table 3 presents the number of concepts per classification for each method. For all the classifications, more concepts are built by methods using the second dataset. The increase is almost 50% for TCA and RCAC and twice as high for RCAR and GFCA-Ternary. Despite the notable difference in the number of plants (30 vs 9), the classifications of RCAG and GFCA-Binary contain two and four additional concepts, respectively. This slight difference results from considering plants as relations between organisms and pests, which increases the number of relational attributes.

Regarding the other methods, the additional concepts originate from the modeling of the 22 new 3-tuples, which implement the protection of Z.mays against the additional pest S.frugi using 22 new plants.

Readability and ease of use. TCA and GFCA both present their results as a single package. TCA lists the triadic concepts, and GFCA connects them in a single graph. RCA also connects the concepts but a switch between lattices is required to access the overall classification. When a classification is used as a support for exploration, classifying the objects per category, as provided by RCAG, is a good solution. For instance, RCAC and RCAG show that A.escu and B.ole are equivalent (CProt8) and share more with Z.mays than with the other species (CProt9). GFCA-Ternary is more difficult to read than GFCA-Binary because it has more concepts and uses ellipse nodes (not shown in this paper) in addition to concept nodes to represent ternary relationships.

Modeling symmetry. TCA, RCAR, and GFCA-Ternary modeling considers the three object categories symmetrically. Thus, 3-tuples sharing characteristics at a given position are recognizable at a glance. RCAG, RCAC, and GFCA-Binary modeling does not consider the three object categories symmetrically. For instance, RCAC classifies protecting plants in relation to targeted pests and protected organisms. Reverse and transitive relationships can be included in the formal contexts to achieve symmetry, but the cost is increased complexity. GFCA-Binary and RCAG handle plants as relation labels instead of objects, implying the absence of plant concepts in the lattice. Two other representations are also possible, where protected organisms or pests are used as relation labels.

Visibility of concept hierarchies and relational patterns. Triadic lattices are difficult to fully visualize and understand, even with the projection paradigm used by FCA Tools Bundle [START_REF] Kis | FCA tools bundle -A tool that enables dyadic and triadic conceptual navigation[END_REF]. RCA clearly depicts the concept hierarchies, one for each object type, but not the relational patterns. GFCA has three output modes: only showing hierarchies such as RCA, only showing relational patterns, or showing the two combined. The latter is the richest representation but it is more difficult to grasp.

Extensibility and limits. All the methods allow the user to return to the original 3-tuples, provided a nominal scaling was used for RCA and GFCA. Like classical FCA, all the implementations of TCA, RCA (with ∃, ∃∀ and ∃⊇), and GFCA are sensitive to noisy data, in that missing 3-tuples increase the number of concepts. For RCA, noisy data can be processed with percentage quantifiers. TCA is restricted to ternary relations: objects, attributes and conditions cannot have their own description. These limitations reduce the modeling options. RCA and GFCA methods can apply different models depending on the expert's needs. Using ∃ quantifier, RCA is less precise than the other FCA extensions, but this allows experts to hypothesize and suggest new protection solutions. As RCA and GFCA are relatively flexible, adding information on objects of the 3-tuples is possible, such as taxonomy for the species. GFCA is less flexible than RCA, as its patterns are exclusively existential, and the relations are two-way. But GFCA patterns can contain cycles while RCA patterns are (possibly infinite) trees, like class expressions in description logics [START_REF]The Description Logic Handbook: Theory, Implementation, and Applications[END_REF].

Conclusion

The Knomana Knowledge Base (KKB) is a typical example of a dataset in the environmental domain, characterized by a lack of information concerning the name of some species and a main ternary relationship. In this paper, we address these two questions using a KKB excerpt. We introduce an aggregation hierarchy on species names and abbreviation spp. We define several representations of the ternary relationship to analyze the KKB excerpt with three FCA extensions. The discussion highlights the complementarity of the different approaches.

In a future study, we plan to deepen the qualitative analysis on pests, starting with Spodoptera species and to define an analytical process that takes advantage of the three FCA extensions. We also plan to explore variants of the representations proposed in this paper. Our long-term objective is to provide domain experts with efficient tools to analyze their dataset, not only to extract existing knowledge in a deductive way, but also to assist them in setting up new hypotheses, and suggesting new experiments for bio-pesticides.

Fig. 1 .

 1 Fig. 1. Aggregation hierarchy used to propagate the spp value in the first dataset.

Fig. 2 .

 2 Fig. 2. RCAC Concept lattices: protected org. (left), plants (middle), pests (right)

Fig. 3 .

 3 Fig. 3. Pests (RCAR) CProt7

  CProt5 Z.mays ∃ A.indi(CPest5) ∃ A.indi(CPest3) ∃ C.spp(CPest5) ∃ C.spp(CPest3) ∃ C.spp(CPest4) ∃ C.spp(CPest6) ∃ C.opu(CPest5) ∃ C.opu(CPest3) ∃ C.opu(CPest4) ∃ C.opu(CPest6) Z.mays CProt9 ∃ A.indi(CPest4) ∃ A.indi(CPest6)

Fig. 4 .

 4 Fig. 4. Data models (left), RCAG lattices: Protected org. (middle), pests (right)

Fig. 5 .

 5 Fig. 5. Concept lattices (excerpt) for GFCA-Ternary (edges as relational attributes)

Table 1 .

 1 Triadic context, with highlighted concept TC5 (in red)

	Z. mays A. escu B. oler G. hirs S. lyco R. com	× × ×	× × × × ×	× ×	×

Table 2 .

 2 Three triadic concepts among the 9 concepts built using Table1

TC2 Protected A.escu B.ole G.hirsu R.com S.lyco Z.mays Plants Pests S.litto S.litto+spp S.litu S.litu+spp S.spp S.spp-any TC5 Protected A.escu B.ole Plants A.indi C.papa Pests S.litto S.litto+spp S.spp-any TC6 Protected A.escu B.ole Z.mays Plants A.indi Pests S.litto+spp S.spp-any

  RCA enables various encodings of datasets. For example, two alternatives can be proposed for the above encoding named RCAC (for RCA with Chain). They

			CProt7
			∃ protectedBy(CPlant10)
		CProt10 ∃ protectedBy(CPlant12)	CProt11 ∃ protectedBy(CPlant13) ∃ protectedBy(CPlant14)				CPlant10 ∃ treats(CPest6)
	CProt6 R.com ∃ protectedBy(CPlant9)	∃ protectedBy(CPlant1) CProt9 ∃ protectedBy(CPlant11)	CProt3 G.hirsu ∃ protectedBy(CPlant3)			CPlant14 ∃ treats(CPest4)	CPlant12 ∃ treats(CPest5)
	R.com				G.hirsu						CPest6
	CProt5 Z.mays ∃ protectedBy(CPlant8) ∃ protectedBy(CPlant7) Z.mays	CProt8 ∃ protectedBy(CPlant2) CPlant3 CPlant4	CProt4 S.lyco ∃ protectedBy(CPlant6) CPlant13 ∃ treats(CPest1) ∃ protectedBy(CPlant5) ∃ protectedBy(CPlant4) CPlant2 CPlant5 CPlant6 S.lyco	CPlant11 ∃ treats(CPest3) CPlant1 CPlant7 ∃ treats(CPest2) CPlant9 S.spp+any W.pro W.pro CPest4 CPest5 CPlant8 S.litto+spp S.litu+spp
			CProt1 A.escu D.dume CProt2 B.ole D.dume	V.cane V.cane	V.fusca V.fusca	V.parvi V.parvi	C.papa C.papa	A.indi A.indi	C.opu C.opu	C.spp CPest1 C.spp S.litto	CPest3 S.spp	CPest2 S.litu
			A.escu	B.ole							S.litto	S.spp	S.litu
								CPlant0
			CProt0 ∃ protectedBy(CPlant0)				∃ treats(CPest0)		CPest0

2 (middle)) groups plants (A. indi, C. papa) that treat at least one pest in CP est1 (S.litto) (relational attribute ∃treats(CP est1)). Concept CP rot8 (Fig.

2

(left)) groups organisms (A.escu, B.ole) that are protected by at least one plant in CP lant2, CP lant1 and CP lant13 (relational attributes ∃protectedBy(CP lant2), ∃protectedBy(CP lant1) and ∃protectedBy(CP lant13), the latter being inherited). This concept chain, which crosses the three lattices, corresponds to Triadic concept TC5. It would also be found with quantifier ∃⊇. Similarly, the Concept chain CP rot9, CP lant1 and CP est6 corresponds to Triadic concept TC6. Triadic concept TC2 does not appear with quantifier ∃, because all protected organisms have at least one protecting plant against one pest: situation (CP o, CP l, CP e) such that ∀po ∈ CP o, ∃pl ∈ CP l, (po, pl) ∈ protectedBy, and ∃pe ∈ CP e, (pl, pe) ∈ treats, that could be summarized as pattern ∃protectedBy∃treats. TC2 would appear with quantifier ∃⊇.

Table 3 .

 3 Classifications obtained by FCA extensions (NA=Not applicable)

		TCA RCAC	RCAR	RCAG	GFCA-Ter. GFCA-Binary
	#concepts 2 nd dataset 12	59	99	18	63	22
	2 nd dataset content (organism-plant-pest)		(12-38-9)	protection 50 (8-32-9)	(10-NA-9) + 30 relations	(12-41-10) (12-NA-10)
	#concepts 1 st dataset	9	33	50	16	36	18

Taken in this paper in the broadest sense to designate bio-aggressors.

Priscilla Keip et al.

https://fca-tools-bundle.com/

http://dataqual.engees.unistra.fr/logiciels/rcaExplore

Source code and user manual at https://bitbucket.org/sebferre/graph-fca/

This work was supported by the French National Research Agency under the Investments for the Future Program, referred as ANR-16-CONV-0004.


Yves Ntilikina
email: yves.ntilikina@chru-strasbourg.fr

Yann Philippe Charles

Sylvain Persohn

Wafa Skalli

Influence of double rods and interbody cages on quasistatic range of motion of the spine after lumbopelvic instrumentation

Keywords: Spinal deformity, Pelvic fixation, Double rod instrumentation, Anterior lumbar interbody fusion, Biomechanics, Lumbosacral range of motion, Rod strain

Purpose This in vitro biomechanical study compares residual lumbar range of motion (ROM) and rod strain after lumbopelvic instrumentation using 2 rods, 4 rods and interbody cages. Methods Seven human cadaveric specimens were instrumented from L1 to sacrum, and pelvic screws were implanted. The pelvis was constrained and moments up to 7.5 Nm were applied to T12. Segmental L1-S1 ROM was analyzed by tracking radiopaque balls implanted in each vertebra using biplanar radiographs. Deformation within principal rods was measured by strain gauges. Four configurations were compared: 2 rods (2R), 4 rods (4R), 4 rods + ALIF at L4-L5 and L5-S1 (4R + ALIF), 2 rods + ALIF (2R + ALIF). Results Intact average global L1-S1 ROM was 42.9° (27.9°-66.0°) in flexion-extension (FE), 35.2° (26.8°-51.8°) in lateral bending (LB), 18.6° (6.7°-47.8°) in axial rotation (AR). In FE, average ROM was 1.9° with both 4-rod configurations versus 2.5° with 2R and 2.8° with 2R + ALIF (p < 0.05). In LB, ROM ranged between 1.2° and 1.5° without significant differences. In AR, ROM was 2.5° with both 4-rod configurations versus 2.9° with 2R (p = 0.07) and 3.1° with 2R ALIF (p = 0.01). In FE, strain decreased by 64% and 65% in principal rods at L3-L4 with 4-rod. When comparing 2-rod configurations, strain decreased by 1% in flexion and increased by 22% in extension at L3-L4 when adding an ALIF at L4-L5 and L5-S1. Conclusions Double rods and interbody cages decrease residual ROM in FE and AT. Double rods seem efficient in limiting strain in principal rods. The use of single rods with cages at the lumbosacral junction increases strain at the first adjacent level without cage.

Introduction

Degenerative lumbar scoliosis and sagittal malalignment represent common spinal deformities in the aging population. Adult spinal deformity (ASD) associated with imbalance has an impact on health-related quality of life (QoL) [START_REF] Smith | Change in classification grade by the SRS-Schwab Adult Spinal Deformity Classification predicts impact on health-related quality of life measures: prospective analysis of operative and nonoperative treatment[END_REF][START_REF] Schwab | International Spine Study Group (ISSG) (2013) Radiographical spinopelvic parameters and disability in the setting of adult spinal deformity: a prospective multicenter analysis[END_REF]. Surgical treatment might be considered if conservative treatment remains inefficient on low back and leg pain, and in progressive trunk imbalance. Although surgical treatment of ASD improves QoL, the incidence of long-term mechanical complications is reported between 30 and 40% [START_REF] Kelly | Fate of the adult revision spinal deformity patient: a single institution experience[END_REF][START_REF] Zhu | Unanticipated revision surgery in adult spinal deformity: an experience with 815 cases at one institution[END_REF]. These rates increase if the spinal deformity and degenerative changes require instrumentation to the sacrum [START_REF] Charosky | Complications and risk factors of primary adult scoliosis surgery: a multicenter study of 306 patients[END_REF][START_REF] Blamoutier | Surgery of lumbar and thoracolumbar scolioses in adults over 50. Morbidity and survival in a multicenter retrospective cohort of 180 patients with a mean follow-up of 4.5 years[END_REF][START_REF] Riouallon | Risk of revision surgery for adult idiopathic scoliosis: a survival analysis of 517 cases over 25 years[END_REF]. Among failures related to long instrumentation including the lumbosacral junction, distal screw loosening represents a common problem. In vitro biomechanical tests have demonstrated that an additional pelvic fixation has the highest potential to protect the S1-anchorage [START_REF] Volkheimer | Is pelvic fixation the only option to provide additional stability to the sacral anchorage in long lumbar instrumentation? A comparative biomechanical study of new techniques[END_REF]. Nevertheless, the use of strong iliac seems to be associated with an increased rate of rod fractures [START_REF] Schwab | Risk factors for major peri-operative complications in adult spinal deformity surgery: a multi-center review of 953 consecutive patients[END_REF][START_REF] Yamato | Committee for Adult Deformity, Japanese Scoliosis Society (2017) A Japanese nationwide multicenter survey on perioperative complications of corrective fusion for elderly patients with adult spinal deformity[END_REF]. Implant failure like rod breakage often indicates the presence of pseudarthrosis, which can lead to revision surgery.

It appears that posterolateral fusion might not be sufficient when instrumenting the thoracolumbar spine including the sacrum and pelvis. A rigid pelvic fixation might lead to fatigue of lumbar instrumentation under cyclic loading (daily activities, walking and recurrent anterior malalignment) which increases the risk of pseudarthrosis and rod fracture. Therefore, an anterior column support and fusion using interbody cages might be recommended to avoid pseudarthrosis [START_REF] Vaz | Bone grafting options for lumbar spine surgery: a review examining clinical efficacy and complications[END_REF][START_REF] Mummaneni | Guideline update for the performance of fusion procedures for degenerative disease of the lumbar spine. Part 11: interbody techniques for lumbar fusion[END_REF][START_REF] Mobbs | Lumbar interbody fusion: techniques, indications and comparison of interbody fusion options including PLIF, TLIF, MI-TLIF, OLIF/ATP, LLIF and ALIF[END_REF]. First clinical studies suggest that multiple rod constructs could reduce the incidence of pseudarthrosis [START_REF] Hyun | Comparison of standard 2-rod constructs to multiple-rod constructs for fixation across 3-column spinal osteotomies[END_REF][START_REF] Merrill | Multirod constructs can prevent rod breakage and pseudarthrosis at the lumbosacral junction in adult spinal deformity[END_REF][START_REF] Daniels | International Spine Study Group (2018) Rod fracture after apparently solid radiographic fusion in adult spinal deformity patients[END_REF][START_REF] Guevara-Villazón | Multiple-rod constructs in adult spinal deformity surgery for pelvic-fixated long instrumentations: an integral matched cohort analysis[END_REF][START_REF] Gupta | A novel 4-rod technique offers potential to reduce rod breakage and pseudarthrosis in pedicle subtraction osteotomies for adult spinal deformity correction[END_REF]. Figure 1 illustrates a clinical case of degenerative lumbar scoliosis instrumented with a 4-rod construct and additional interbody cages.

The main hypothesis of this study is that the use of interbody cages and/or multiple rods could decrease residual minimal range of motion in the instrumented segment, which might lead to nonunion and implant failure on long term.

The purpose of this biomechanical study was to investigate residual mobilities within the entire lumbar spine and strain in the rods after lumbosacral and pelvic instrumentation using single rods, double rods and interbody cages.

Materials and methods

Specimens

Seven fresh-frozen human cadaveric T12-pelvis specimens were tested. Specimens without spinal pathologies that would influence biomechanical testing (osteoporosis, fractures, metastases, major decrease in intervertebral disk height or facet joint osteoarthritis) were selected by visual examination and computed tomography (CT). All donors were males, and their average age was 57.7 (42-63) years. The average bone mineral density was 104.2 mg/cm 3 .

The specimens were freshly dissected, sealed in double plastic bags, frozen and stored at -20 °C until testing. The specimens were thawed to 6 °C for 12-14 h before starting the preparation process. The experiment itself was performed at room temperature. Soft tissues were removed, leaving all ligaments, joint capsules, disks and bony structures intact.

After reconstruction of the specimen's geometry from three-dimensional (3D) CT images (Medical imaging interaction toolkit, MITK 2016.11, Heidelberg, Germany), specimen-specific custom-made molds were 3D-printed to constrain the T12 vertebral body and spinous process cranially and the ischium caudally. The bone-mold interface was additionally secured with small screws. Two-millimeter-diameter stainless steel balls were then implanted in the cortical bone from L1 to S1 after drilling a small pilot hole (one in each spinous process and four in each vertebral body). Six balls were implanted in the sacrum and three in each iliac wing. Figure 2 shows anterior-posterior and lateral radiographs (EOS™, EOS Imaging, Paris, France) that allowed calculating 3D positions of the balls using MATLAB software (MathWorks, Natick, MA, USA). The coordinate systems for each group of balls (RBi) were associated with the respective vertebrae [START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF], the sacrum and pelvis [START_REF] Mitton | 3D reconstruction of the pelvis from bi-planar radiography[END_REF][START_REF] Ghostine | Influence of patient axial malpositioning on the trueness and precision of pelvic parameters obtained from 3D reconstructions based on biplanar radiographs[END_REF][START_REF] Muth-Seng | Experimental analysis of the lower cervical spine in flexion with a focus on facet tracking[END_REF]. The global coordinate system of the EOS cabin was defined as R0. The transformation matrices for each RBi to the global R0 allowed calculating relative successive positions between vertebrae and sacrum.

Kinematic testing

The flexibility tests were conducted in a specific spine-testing device. The pelvic 3D-printed mold was rigidly fixed to the pelvis and to a frame in the EOS™ system. The cranial mold, fixed on T12, was mounted to a motor (Gearmotor 3242G024C, Faulhaber SAS, Montigny-le-Bretonneux, France), which was unconstrained in the three directions of translation and two perpendicular axes of rotation (Fig. 3). This system allowed applying quasistatic moments for flexion-extension, lateral bending and axial torsion by gradual loading in steps of 1.5 Nm, with an interval of 15 s between each step, until a maximum of 7.5 Nm was reached [START_REF] Wilke | Testing criteria for spinal implants: recommendations for the standardization of in vitro stability testing of spinal implants[END_REF]. Three preloading cycles were performed to compensate for small viscoelastic effects. A 6-axis load cell (FTD-Gamma SI-130-10, ATI Industrial Automation, Apex, NC, USA) placed under the specimen allowed checking whether pure moments were applied. After three preconditioning cycles, flexion and extension were completed during the same loading-unloading cycle. The same applied for right and left lateral bending as well as right and left axial torsion. Biplanar radiographs were taken at each increment, to track the metal balls. The successive positions of these markers were measured during the loading-unloading cycle values, which allowed to determine three angular displacements (rotations around X, Y and Z axis) and three linear displacements 3 Intact specimen mounted in the test bench in the biplanar X-rays system (white star) with the motor linked to the T12 mold by a universal joint (white arrow), the 6-axis sensor below the specimen (black arrow), the balancing weight above the motor (black star) (translations along TX, TY and TZ). The centers of the metal markers were localized automatically with templatematching using normalized cross-correlation. The incertitude (95% confidence interval) of this motion tracking method was estimated using a motorized linear translation stage and was below 0.049 mm for each axis.

Load-displacement curves were obtained, thus quantifying 3D segmental range of motion (ROM) between vertebrae and sacrum for the noninstrumented specimen. The global ROM of L1 related to the sacrum was considered for instrumented configurations.

The uncertainty of mobility measurement was estimated using a Monte Carlo method introducing Gaussian noise to the different sources of possible errors and performing 300 simulations. The confidence interval (95% CI) was lower than 0.12° for the main rotations, 0.23° for coupled rotations and 0.12 mm for linear displacements.

Instrumentation configurations

An orthopedic spine surgeon performed all instrumentations (Erisma Deformity ® , Clariance, Beaurains, France) using the free-hand technique for pedicle screw placement. Polyaxial pedicle screws were placed bilaterally from L1 to L5 (6.5 mm diameter, 45 mm length) and at S1 (7.5 mm diameter, 45 mm length). Iliac screws (8.5 mm diameter, 70 mm length) were placed bilaterally in a subcrestal fashion. Cobalt-chromium rods (CoCr, 5.5 mm diameter) were contoured and connected to the screws while taking care to minimize the need for reduction techniques. Offset connectors between iliac screws and rods were utilized. A crosslink connector was placed between L5 and S1 screws. Two uniaxial strain gauges (KFG-2N-120-C1-11 L3M2R, KYOWA, Tokyo, Japan) were stuck on the posterior side of each rod midway between L3 and L4 screws (Fig. 4). This positioning allowed rod strain measurements in flexion-extension.

This standard setting of posterior L1-pelvis instrumentation was defined as 2-rod configuration (2R). Afterward, two additional 5.5-mm-diameter CoCr rods were placed medially and connected to the principal rods between L1 and L2 and between the S1 and iliac screws using side-by-side connectors, which was defined as 4-rod configuration (4R). An anterior discectomy was then performed at L4-L5 and L5-S1 while preserving the endplates and the posterior longitudinal ligament. Anterior lumbar interbody fusion (ALIF) polyether-ether-ketone (PEEK) cages were then inserted at both levels. The cage height (10-14 mm) was adapted to each level in order to fill out the interbody space (4R + ALIF configuration). The accessory rods were then removed for the last step, which represented a 2-rod construct with L4-L5 and L5-S1 ALIF cages (2R + ALIF configuration). Figure 5 shows radiographs for each configuration.

Segmental ROM was measured between L1 and S1 in flexion and extension, left and right lateral bending and axial torsion for each instrumentation configuration. Configurations were then compared to the intact specimen, and results were reported as percentage using the formula: variation = ROM instrumented -ROM intact/ROM intact × 100. The strain in the principal rods was measured by strain gauges for each configuration between L3 and L4 in flexion-extension only.

Statistical analysis

Data were analyzed using R software (R Foundation, Vienna, Austria). Quantitative data were expressed as average, standard deviation and range. A Shapiro-Wilk test was used to check normal distribution. A linear mixed model was then used to analyze quantitative variables with a fixed effect (configuration) and a random effect (specimen). ROM was compared between the intact specimen and each instrumented configuration. Rod strain was compared between instrumented configurations. The significance level was set at p < 0.05.

Results

Segmental lumbar range of motion

Global lumbar range of motion with instrumentation

Table 3 demonstrates ROM of the entire L1-S1 segment for the intact specimen and the four instrumented configurations.

When comparing the intact specimen to instrumented configurations in flexion-extension, the average ROM decreased by 94% with 2 rods (2R and 2R + ALIF) and by 96% with 4 rods (4R and 4R + ALIF, respectively). The decrease in ROM was significant for each configuration (p < 0.0001). The average ROM was 2.5° with 2R and 2.8° with 2R + ALIF, whereas ROM was 1.9° with 4R and 4R + ALIF, respectively. When comparing the influence of the number of rods, 4-rod configurations reduced the ROM slightly but significantly compared to 2-rod configurations: 2R versus 4R (p = 0.035), 4R versus 2R + ALIF (p = 0.010), 4R + ALIF versus 2R + ALIF (p = 0.002). There was no significant difference between 2R and 2R + ALIF (p = 0.444) and between 4R and 4R + ALIF (p = 0.756).

In lateral bending, average ROM ranged between 1.2° and 1.5° and each configuration reduced ROM by 96% compared to the intact spine (p < 0.0001). There was no significant difference between the instrumented configurations.

In axial torsion, average ROM was decreased by 84% with 2R, by 87% with 4R and 4R + ALIF, respectively, and by 83% with 2R + ALIF compared to the intact spine (p < 0.02). When comparing the influence of number of rods and additional cage implantation, it appeared that 4 rods would mainly influence ROM: 2R versus 4R (p = 0.066), 4R versus 2R + ALIF (p = 0.014), 4R + ALIF versus 2R + ALIF (p = 0.011). There was no significant difference between 2R

Rod strain

Table 4 demonstrates average rod strain (RS) in the principal rods at the level L3-L4. There was no difference between left and right RS in any configuration. When comparing 2-and 4-rod configurations in extension, 4R and 4R + ALIF decreased strain by 64% in the principal rods compared to 2R, respectively. In flexion, strain decreased by 68% with 4R and 65% with 4R + ALIF. For flexion and extension, the differences were significant between 2-rod and 4-rod configurations: 2R versus 4R (p = 0.016), 2R versus 4R + ALIF (p = 0.015), 2R + ALIF versus 4R (p = 0.011), 2R + ALIF versus 4R + ALIF (p = 0.015).

When analyzing the influence of cages with 2-rod configurations, it appeared that strain decreased by 1% in flexion with 2R + ALIF compared to 2R. In extension, cage implantation at L4-L5 and L5-S1 (2R + ALIF) increased strain by 22% at L3-L4 compared to 2R.

Discussion

Pseudarthrosis and rod fractures remain a major challenge in adult spinal deformity surgery since instrumentation to the sacrum and pelvis is often required. Daniels et al. [START_REF] Daniels | International Spine Study Group (2018) Rod fracture after apparently solid radiographic fusion in adult spinal deformity patients[END_REF] and Lertudomphonwanit et al. [START_REF] Lertudomphonwanit | Rod fracture in adult spinal deformity surgery fused to the sacrum: prevalence, risk factors, and impact on health-related quality of life in 526 patients[END_REF] analyzed risk factors for rod fractures and demonstrated that the patient's age, obesity, residual anterior malalignment, the number of instrumented levels, the use of osteotomies, the rod diameter and alloy might play a role. Four-rod instrumentation has been used recently to prevent pseudarthrosis and implant failure. Hyun et al. and Gupta et al. [START_REF] Gupta | A novel 4-rod technique offers potential to reduce rod breakage and pseudarthrosis in pedicle subtraction osteotomies for adult spinal deformity correction[END_REF] compared radiologic outcomes after 2-rod and 4-rod instrumentation. These first clinical studies indicate that the incidence of pseudarthrosis is lower with 4-rod constructs and that revision surgery for rod fractures was not required. However, long-term results are required to better understand the effects of this new instrumentation strategy. It seems that satellite rods limit minimal ROM and strain within principal rods, which could prevent rod fractures. Combining lumbosacral and pelvic instrumentation with interbody cages represents an additional strategy to prevent mechanical complications [START_REF] Barbera | Preclinical evaluation of posterior spine stabilization devices: can the current standards represent basic everyday life activities?[END_REF][START_REF] Barbera | Preclinical evaluation of posterior spine stabilization devices: can we compare in vitro and in vivo loads on the instrumentation?[END_REF]. Furthermore, Banno et al. [START_REF] Banno | Multi-rod constructs can increase the incidence of iliac screw loosening after surgery for adult spinal deformity[END_REF] investigated the incidence of proximal pedicle and iliac screw loosening with 2-rod versus 4-rod constructs. A radiologic halo developed around screws in 21% of patients with 2-rod versus 33% after 4-rod instrumentation. This indicates that more rigid multiple rod instrumentations might lead to enhanced stress at the proximal and distal end of the construct. First studies on multiple rods focused on pedicle subtraction osteotomies (PSO). However, the exact role of additional cages remained unclear, and analyzing the scenario of adult scoliosis instrumentation without three-column osteotomies seemed mandatory.

Three-dimensional segmental ROM has been analyzed in vivo on lumbar spine radiographs of healthy volunteers [START_REF] Pearcy | Three-dimensional X-ray analysis of normal movement in the lumbar spine[END_REF][START_REF] Pearcy | Axial rotation and lateral bending in the normal lumbar spine measured by three-dimensional radiography[END_REF][START_REF] Dvorák | Functional radiographic diagnosis of the lumbar spine. Flexion-extension and lateral bending[END_REF][START_REF] Hayes | Roentgenographic evaluation of lumbar spine flexion-extension in asymptomatic individuals[END_REF]. These studies have demonstrated that the amount of ROM depends on the lumbar level. The highest ROM value in FE is located at L4-L5 and L5-S1. The largest ROM in LB and AT is located at L2-L3 and L3-L4. Panjabi et al. [START_REF] Panjabi | Mechanical behavior of the human lumbar and lumbosacral spine as shown by three-dimensional load-displacement curves[END_REF] studied segmental ROM from L1-L2 to L5-S1 in vitro and demonstrated results that were concordant with previous clinical radiographic findings. Guan et al. [START_REF] Guan | Moment-rotation responses of the human lumbosacral spinal column[END_REF], Heuer et al. [START_REF] Heuer | Stepwise reduction of functional spinal structures increase range of motion and change lordosis angle[END_REF] and Charles et al. [START_REF] Charles | Influence of an auxiliary facet system on lumbar spine biomechanics[END_REF] studied L4-L5 ROM in vitro, and Hlubek et al. [START_REF] Hlubek | Iliac screws may not be necessary in long-segment constructs with L5-S1 anterior lumbar interbody fusion: cadaveric study of stability and instrumentation strain[END_REF] reported normal values for L5-S1 ROM. Our results were in line with these previous findings, showing similar ROM corridors, when loading the entire lumbar spine up to 7.5 Nm.

Experimental cadaveric studies analyzing 4-rod posterior instrumentation can be found in the recent literature. However, most studies are based on PSO models [START_REF] Hlubek | Iliac screws may not be necessary in long-segment constructs with L5-S1 anterior lumbar interbody fusion: cadaveric study of stability and instrumentation strain[END_REF][START_REF] Scheer | Biomechanical analysis of revision strategies for rod fracture in pedicle subtraction osteotomy[END_REF][START_REF] Hallager | Use of supplemental short pre-contoured accessory rods and cobalt chrome alloy posterior rods reduces primary rod strain and range of motion across the pedicle subtraction osteotomy level: an in vitro biomechanical study[END_REF][START_REF] Dahl | Kinematic efficacy of supplemental anterior lumbar interbody fusion at lumbosacral levels in thoracolumbosacral deformity correction with and without pedicle subtraction osteotomy at L3: an in vitro cadaveric study[END_REF][START_REF] Barbera | Biomechanical advantages of supplemental accessory and satellite rods with and without interbody cages implantation for the stabilization of pedicle subtraction osteotomy[END_REF][START_REF] Barbera | Biomechanical in vitro comparison between anterior column realignment and pedicle subtraction osteotomy for severe sagittal imbalance correction[END_REF]. Scheer et al. [START_REF] Scheer | Biomechanical analysis of revision strategies for rod fracture in pedicle subtraction osteotomy[END_REF] compared multidirectional bending stiffness of multiple rod instrumentation with iliac fixation and L3 PSO in seven human spines (T11-pelvis). Satellite rods restored stiffness in FE and LB, but not in AT. The addition Strain is an index that can be used to predict rod failure. Hallager et al. [START_REF] Hallager | Use of supplemental short pre-contoured accessory rods and cobalt chrome alloy posterior rods reduces primary rod strain and range of motion across the pedicle subtraction osteotomy level: an in vitro biomechanical study[END_REF] used uniaxial strain gauges and reported that accessory rods and CoCr rods reduced relative surface strain on primary rods, irrespective of the construct. The use of four CoCr rods and interbody cages provided 48% reduction in primary rod strain compared to 2-rod constructs. They reported a small decrease in primary and accessory rod strain at PSO level when adding interbody cages. Furthermore, strain was lower with CoCr compared to titanium rods. La Barbera et al. [START_REF] Barbera | Biomechanical advantages of supplemental accessory and satellite rods with and without interbody cages implantation for the stabilization of pedicle subtraction osteotomy[END_REF] used rosette gauges in a PSO model. When comparing 4-rod and 2-rod constructs, primary rod strain decreased by 74.4% in FE, 90.6% in LB and 63.5% in AR. This effect was enhanced when adding interbody cages. La Barbera et al. [START_REF] Barbera | Biomechanical in vitro comparison between anterior column realignment and pedicle subtraction osteotomy for severe sagittal imbalance correction[END_REF] described a similar effect in an anterior column realignment (ACR) model. The role of 4-rod constructs and interbody cages in reducing primary rod strain after PSO was further confirmed in finite element models [START_REF] Vosoughi | Optimal satellite rod constructs to mitigate rod failure following pedicle subtraction osteotomy (PSO): a finite element study[END_REF][START_REF] Luca | Anterior support reduces the stresses on the posterior instrumentation after pedicle subtraction osteotomy: a finite-element study[END_REF][START_REF] Luca | Instrumentation failure following pedicle subtraction osteotomy: the role of rod material, diameter, and multi-rod constructs[END_REF]. Although the findings of these experimental settings focus on three-column osteotomies, the findings are in agreement with the results of our study, which represented a setting of lumbopelvic instrumentation for adult scoliosis. Godzik et al. [START_REF] Godzik | Supplemental rods are needed to maximally reduce rod strain across the lumbosacral junction with TLIF but not ALIF in long constructs[END_REF] demonstrated that strain would increase in the caudal part of the construct when instrumenting the pelvis. Kleck et al. [START_REF] Kleck | Strain in posterior instrumentation resulted by different combinations of posterior and anterior devices for long spine fusion constructs[END_REF][START_REF] Kleck | Reply to Letter to Editor: strain in posterior instrumentation resulted by different combinations of posterior and anterior devices for long spine fusion constructs[END_REF] demonstrated that spinopelvic fixation increases L5-S1 rod strain especially during flexion. The addition of an ALIF divided strain by two in FE, but it had no effect in AT. Moreover, Hlubek et al. [START_REF] Hlubek | Iliac screws may not be necessary in long-segment constructs with L5-S1 anterior lumbar interbody fusion: cadaveric study of stability and instrumentation strain[END_REF] investigated the effect of L5-S1 ALIF and transforaminal lumbar interbody fusion (TLIF) cages as well as iliac screws on rod and S1 pedicle screw strain. They demonstrated that iliac screws were protective of sacral screws but increased rod strain at L5-S1. When using iliac screws with 2 rods, the ALIF reduces rod strain significantly, whereas the TLIF did not. In our study strain gauges were placed at L3-L4, which represented the first level adjacent to a 360° instrumentation (ALIF L4-L5 and L5-S1). Our findings suggested that the use of 2 rods with cages could increase rod strain in extension at the first adjacent level without cage. This is in line with the clinical findings of Lertudomphonwanit et al. [START_REF] Lertudomphonwanit | Rod fracture in adult spinal deformity surgery fused to the sacrum: prevalence, risk factors, and impact on health-related quality of life in 526 patients[END_REF] who showed that 44% of rod fractures occurred at L3-L4 when cages were implanted at L4-L5 and L5-S1. Our results indicated that 4 rods significantly decreased strain in primary rods at the adjacent level.

Although our findings provide a foundation for future clinical research with 4-rod constructs, some limitations are inherent to biomechanical studies. Standard lumbar loading conditions using pure moments up to 7.5 Nm were applied [START_REF] Wilke | Testing criteria for spinal implants: recommendations for the standardization of in vitro stability testing of spinal implants[END_REF], although loading conditions for the entire lumbosacral spine might be revised for longer instrumentation configurations. Testing focused on the lumbosacral spine only, whereas scoliosis instrumentation usually involves thoracic segments as well. Moreover, rod strain measurement was analyzed at a single location in FE because of the use of uniaxial strain gauges, even if maximal strain and stress values could be located elsewhere. An additional analysis in LB and AR should be considered using rosette strain gauges in future studies. Nevertheless, such analysis with ROM and rod strain quantitative assessment provide a solid base for further finite element modeling and validation, thus yielding a deeper stress analysis in the entire spine construct to further complete the present results.

Conclusion

The results of this study indicate that using double rods and interbody cages can decrease residual ROM in FE and AT. The use of 4-rod constructs seemed to be the most efficient method in limiting strain in principal rods. The use of 2 rods with ALIF cages at L4-L5 and L5-S1 increases strain at L3-L4.

Fig. 1

 1 Fig. 1 Preoperative posterior-anterior (a) and lateral (b) full spine standard radiographs of a patient with adult scoliosis and postoperative posterior-anterior (a) and lateral (b) radiographs after T5-ilium instrumentation using double rods and interbody fusion cages

Fig. 2

 2 Fig. 2 Anteroposterior (a) and lateral (b) biplanar radiographs of an intact specimen showing cranial and caudal custom molds (black arrows) and metallic balls in each vertebra from L1 to S1, the sacrum and pelvis

Fig. 4

 4 Fig.[START_REF] Zhu | Unanticipated revision surgery in adult spinal deformity: an experience with 815 cases at one institution[END_REF] Posterior view of a specimen with 4-rod instrumentation and custom mold fixation (arrows) at T12 and the pelvis (a). Focus on strain gauges (white circles) placed on the principal rods between the pedicle screws of L3 and L4 (b)

Fig. 5

 5 Fig. 5 Biplanar radiographs of the four instrumented configurations: 2 rods (a, b), 4 rods (c, d), 4 rods + ALIF L4-L5 and L5-S1 (e, f), 2 rods + ALIF L4-L5 and L5-S1 (g, h)

Table 1

 1 demonstrates segmental ROM which was measured when a 7.5 Nm bending moment was applied to the intact lumbar specimens. ROM ranged from 7.4° to 10.5° in flexion-extension, 6.0° to 10.3° in lateral bending and 2.0° to 5.8° in axial torsion. The largest ROM was found at the level L4-L5 for each direction.The segmental ROM with instrumentation is descripted in

Table 2 .

 2

Table 1

 1 Segmental lumbar range of motion in intact specimens: average values ± standard deviation (minimum-maximum)

	Level	Flexion-extension	Lateral bending	Axial rotation
	L1-L2	7.4° ± 1.7°(	7.9° ± 2.1°(	2.8° ± 1.4°(
		4.4°-9.6°)	6.0°-11.3°)	1.2°-4.6°)
	L2-L3	7.7° ± 2.8°(	9.5° ± 2.7°(	4.9° ± 3.7°(
		4.1°-13°)	6.7°-13.1°)	1.1°-11.9°)
	L3-L4	7.4° ± 1.8°(	9.7° ± 1.8°(	5.0° ± 3.0°(
		5.0°-10.7°)	8.0°-11.8°)	1.2°-10.4°)
	L4-L5	10.2° ± 3.6°(	10.3° ± 2.9°(	5.8° ± 5.3°(
		4.9°-15.7°)	5.7°-13.8°)	1.2-10.4)
	L5-S1	10.1° ± 6.0°(	6.0° ± 3.8°(	2.0° ± 2.3°(
		2.6°-17.6°)	1.1°-10.2°)	0.6°-7.0°)

Table 2

 2

	Segmental lumbar
	range of motion in instrumented
	specimens: average
	values ± standard deviation
	(minimum-maximum)

Table 3

 3

	Global lumbar (L1-S1) range of motion for the intact spine
	and four instrumented configurations: average values ± standard devi-
	ation (minimum-maximum)
	Configuration Flexion-extension Lateral bending Axial rotation
	Intact spine	42.9° ± 3.1°(	35.2° ± 3.2°(	18.6° ± 6.3°(
		27.9°-66.0°)	26.8°-51.8°)	6.7°-47.8°)
	2 rods	2.5° ± 0.6°( 1.2°-3.7°)* , †	1.2° ± 0.1°( 1.2°-4.6°)*	2.9° ± 0.1°( 1.7°-5.8°) #
	4 rods	1.9° ± 0.3°( 1.1°-3.2°)* , ‡	1.4° ± 0.2°( 0.7°-2.9°)*	2.5° ± 0.3°( 1.5°-4.5°) #, ‡
	4 rods + ALIF 1.9° ± 0.1°( 1.5°-2.9°)* , ‡	1.3° ± 0.2°( 0.6°-3.2°)*	2.5° ± 0.2°( 1.7°-4.4°) #, ‡
	2 rods + ALIF 2.8° ± 0.3°( 1.6°-4.1°)* , †	1.5° ± 0.2°( 0.6°-3.5°)*	3.1° ± 0.6°( 2.0°-6.0°) #, †
	*p < 0.0001 compared to intact
	# p < 0.05 compared to intact
	† p < 0.05 compared to 4 rods
	‡ p < 0.05 compared to 2 rods + ALIF

Table 4 Maximum

 4

	rod strain on the left and the right principal	Configuration Left		Right
	rods for each instrumentation		Extension	Flexion	Extension	Flexion
	in extension (-7.5 Nm) and flexion (7.5 Nm): average values ± standard deviation	2 rods	-0.026 ± 0.010 (-0.047 to -0.014)	0.027 ± 0.010 (0.017 to 0.047)	-0.025 ± 0.009 (-0.049 to -0.014)	0.027 ± 0.010 (0.014 to 0.054)
	(minimum-maximum)	4 rods	-0.009 ± 0.003 (-0.012 to -0.007) *, †	0.009 ± 0.003 (0.006 to 0.013) *, †	-0.009 ± 0.003 (-0.013 to -0.005) *, †	0.009 ± 0.003 (0.006 to 0.013) *, †
		4 rods + ALIF -0.010 ± 0.003 (-0.013 to -0.007) *, †	0.009 ± 0.003 (0.008 to 0.013) *, †	-0.009 ± 0.003 (-0.012 to -0.004) *, †	0.009 ± 0.003 (0.005 to 0.014) *, †
		2 rods + ALIF -0.032 ± 0.012	0.027 ± 0.010	-0.030 ± 0.011	0.027 ± 0.010
			(-0.054 to -0.015)	(0.015 to 0.036)	(-0.058 to -0.013)	(0.013 to 0.039)
		*p < 0.05 compared to 2 rods
		† p < 0.05 compared to 2 rods + ALIF

Service de Chirurgie du Rachis, Hôpitaux Universitaires de Strasbourg, Strasbourg, France

Acknowledgements

The authors would like to acknowlege Clariance Spine for technical and financial support and Société Française de Chirurgie du Rachis (SFCR) for research grant.

Funding A research grant was obtained from the Société Française de Chirurgie Rachidienne (SFCR). Technical and financial support was provided by Clariance.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.