2.1 A brief introduction to description logics

In this section, we recall the basics of DLs that are useful and necessary for understanding the DL expressions given in the paper. This short presentation is based on the DL Handbook (Baader et al., Reference Baader, Calvanese, McGuinness, Nardi and Patel-Schneider2003) and on another more recent introduction to DLs (Krötzsch et al., Reference Krötzsch, Simancik and Horrocks2012).

DLs are a family of knowledge representation languages that are widely used for designing knowledge-based systems. In DLs, there are three kinds of entities: concepts, roles, and individual names. Concepts denote sets of individuals (classes in OWL), roles denote binary relations between the individuals (properties in OWL), and individual names denote simple individuals in the domain. Concepts correspond to unary predicates in first-order logic, roles to binary predicates and individuals to constants. Two levels of knowledge are taken into account: conceptual knowledge (‘Tbox’) including introductions of concepts and roles, and assertional knowledge (‘Abox’) including expressions involving individuals. DLs are equipped with a formal logic-based semantics, and DL systems provide their users with various inference capabilities mainly based on satisfiability, subsumption, and classification of concepts and of individuals.

Elementary descriptions are atomic concepts and atomic roles, while complex descriptions can be built using concept contructors. More precisely, let C be a set of concept names and R be a set of role names disjoint from C. For the so-called $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ description logic, the set of concepts over C and R is inductively defined as follows:

• • Every concept name is an $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ concept description.

• • ⊤ (top) and ⊥ (bottom) are $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ concept descriptions.

• • If C and D are $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ concept descriptions and r is a role name, then the following are also $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ concept descriptions:

  • – C⊓D (conjunction)

  • – C $$\[--><$>\sqcup<$><!--$$ D (disjunction)

  • – ¬C (complete negation)

  • – ∃r.C (existential restriction)

  • – ∀r.C (universal restriction)

Other concept constructions can be defined such as: cardinality restrictions with (≥n r) (‘at least’ restriction) and (≤n r) (‘at most’ restriction), and set of individuals (nominals). Role constructions can be used for extending $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ , such as conjunctions of roles, inverse roles, role hierarchy, etc. Each $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ extension has a name: e.g. $$\[--><$>{\cal A}{\cal L}{\cal C}{\cal N} <$><!--$$ for cardinality restrictions, $$\[--><$>{\cal A}{\cal L}{\cal C}{\cal O} <$><!--$$ for nominals, $$\[--><$>{\cal A}{\cal L}{\cal C}{\cal R} <$><!--$$ for conjunctions of roles, $$\[--><$> {\cal A}{\cal L}{\cal C}{\cal I} <$><!--$$ for inverse roles, and finally $$\[--><$> {\cal A}{\cal L}{\cal C}{\cal H} <$><!--$$ for role hierarchy. Below, the $$\[--><$> {\cal S} <$><!--$$ notation is used where $$\[--><$> {\cal S} <$><!--$$ stands for $$\[--><$>{\cal A}{\cal L}{\cal C} <$><!--$$ and thus $$\[--><$> {\cal S}{\cal H}{\cal O}{\cal I}{\cal N} <$><!--$$ stands for $$\[--><$> {\cal A}{\cal L}{\cal C}{\cal H}{\cal O}{\cal I}{\cal N} <$><!--$$ . Moreover, in $$\[--><$> {\cal S}{\cal H}{\cal O}{\cal I}{\cal N}\!\left( {\cal D} \right) <$><!--$$ , $$\[--><$> {\cal D} <$><!--$$ indicates a concrete domain and this is explained in the Section 2.2.

An interpretation $$\[--><$> {\cal I}\, = \,({{{\rm{\rDelta }}}^{\cal I}} {{,\,.}^{\cal I}} ) <$><!--$$ is based on an interpretation domain $$\[--><$>{{\rDelta }^{\cal I}} <$><!--$$ and on an interpretation function $$\[--><$>{{.}^{\cal I}} <$><!--$$ mapping a concept onto a subset of $$\[--><$>{{\rDelta }^{\cal I}} <$><!--$$ , and a role onto a subset of the product $$\[--><$>{{{\rm{\rDelta }}}^{\cal I}} \times {{{\rm{\rDelta }}}^{\cal I}} <$><!--$$ , with respect to the following equations (where card(X) returns the cardinality of set X).

$$\[--><$$l>\quad\qquad \matrix{ {{{ \top }^{\cal I}} \, = \,{{\rDelta }^{\cal I}} } \hfill \cr \\ {{{ \bot }^{\cal I}} \, = \,\emptyset} \hfill \cr \\ {{{{\left( {{\rm{\tf="courier" C}}\, \sqcap \,{\rm{\tf="courier" D}}} \right)}}^{\cal I}} \, = \,{{{\rm{\tf="courier" C}}}^{\cal I}} \, \cap \,{{{\rm{\tf="courier" D}}}^{\cal I}} } \hfill \cr \\ {{{{\left( {{\rm{\tf="courier" C}}\, \sqcup \,{\rm{\tf="courier" D}}} \right)}}^{\cal I}} \, = \,{{{\rm{\tf="courier" C}}}^{\cal I}} \, \cup \,{{{\rm{\tf="courier" D}}}^{\cal I}} } \hfill \cr \\ {{{{\left( {\neg {\rm{\tf="courier" C}}} \right)}}^{\cal I}} \, = \,{{\rDelta }^{\cal I}} \,\backslash \,{{{\rm{\tf="courier" C}}}^{\cal I}} } \hfill \cr \\ {{{{\left( {\forall {\rm{\tf="courier" r}}.{\rm{\tf="courier" C}}} \right)}}^{\cal I}} \, = \,\left\{ {x\, \in \,{{\rDelta }^{\cal I}} \,|\,\forall y:\left( {x,y} \right)\, \in {{{\rm{\tf="courier" r}}}^{\cal I}} \, \rightarrow \,y\, \in \,{{{\rm{\tf="courier" C}}}^{\cal I}} } \right\}} \hfill \cr \\ {{{{\left( {\exists {\rm{\tf="courier" r}}.{\rm{\tf="courier" C}}} \right)}}^{\cal I}} \, = \,\left\{ {x\, \in \,{{\rDelta }^{\cal I}} \,|\,\exists y:\left( {x,y} \right)\, \in \,{{{\rm{\tf="courier" r}}}^{\cal I}} \, \wedge \,y\, \in \,{{{\rm{\tf="courier" C}}}^{\cal I}} } \right\}} \hfill \cr \\ {{{{\left( {\geq \!{\rm{\tf="courier" m}}\,{\rm{\tf="courier" r}}} \right)}}^{\cal I}} \, = \,\left\{ {x\, \in \,{{\rDelta }^{\cal I}} \,|\,card\left( {\left\{ {y\, \in \,{{\rDelta }^{\cal I}} /\left( {x,y} \right)\, \in \,{{{\rm{\tf="courier" r}}}^{\cal I}} } \right\}} \right)\geq m} \right\}} \hfill \cr \\ {{{{\left( {\leq \!{\rm{\tf="courier" n}}\,{\rm{\tf="courier" r}}} \right)}}^{\cal I}} \, = \,\left\{ {x\, \in \,{{\rDelta }^{\cal I}} \,|\,card\left( {\left\{ {y\, \in \,{{\rDelta }^{\cal I}} /\left( {x,y} \right)\, \in \,{{{\rm{\tf="courier" r}}}^{\cal I}} } \right\}} \right)\leq n} \right\}} \hfill \cr \\ \\ \\ $$

Then, a concept C is satisfiable iff there exists an interpretation $$\[--><$>{\cal I} = \left( {{{\rDelta }^{\cal I}}, {{.}^{\cal I}} } \right) <$><!--$$ such as $$\[--><$>{{{\rm{C}}}^{\cal I}} \ne \emptyset\<$><!--$$ , otherwise C is not satisfiable. A concept C is subsumed by a concept D iff for all interpretation $$\[--><$>{\cal I} <$><!--$$ , $$\[--><$>{{{\rm{\tf="courier" C}}}^{\cal I}} \subseteq {{{\rm{\tf="courier" D}}}^{\cal I}} <$><!--$$ . Satisfiability and subsumption are the base of DL-based reasoning modes. Examples are proposed in Section 2.4.

Hereafter, the usual syntax and semantics of DLs, introduced in Baader et al. (Reference Baader, Calvanese, McGuinness, Nardi and Patel-Schneider2003) as well as the naming conventions and recommendations are respected: that is, a concept name such as MultifocalTumor has a capitalized first letter, an instance name such as PAT has every letters capitalized, and a role name such as hasRecommendation has capitalized first letters except for the first one.

2.2 Description logics and Web ontology language description logic

The knowledge associated with a protocol is represented within OWL DL 1.0, which can be seen as an expressive DL such as $$\[--><$> {\cal S}{\cal H}{\cal O}{\cal I}{\cal N}\left( {\cal D} \right) <$><!--$$ , integrated within standard technologies of the Web such as XML and RDF(S) (Horrocks et al., Reference Horrocks, Patel-Schneider and van Harmelen2003). The notion of an ontology is considered here as an equivalent for a DL knowledge base.

In addition, the system described here makes use of concrete domain within the DL formalism (Lutz et al., Reference Lutz, Areces, Horrocks and Sattler2005; Lutz & Milicic, Reference Lutz and Milicic2007). Following the notations in Lutz (Reference Lutz2003), a concrete domain is a pair $$\[--><$>\left( {{{\rDelta }^{\cal D}}, \,{{\rPhi }^{\cal D}} } \right) <$><!--$$ where $$\[--><$>{{\rDelta }^{\cal D}} <$><!--$$ is a set and $$\[--><$>{{\rPhi }^{\cal D}} <$><!--$$ is a set of predicate names on $$\[--><$>{{\rDelta }^{\cal D}} <$><!--$$ . Here, the concrete domain $$\[--><$>{\cal D} = \left( {{\Bbb R}{\rm{,}}\,{{\rPhi }^{\Bbb R}} } \right) <$><!--$$ is considered, where $$\[--><$>{\Bbb R} <$><!--$$ denotes real numbers, and $$\[--><$>{{\rPhi }^{\Bbb R}} <$><!--$$ is the set of predicate names for comparing real numbers: $$\[--><$> {{\{ \lt }_a},{{\leq }_a}{{, = }_a},{{ \ne }_a},{{\geq }_a}{{, \gt }_a}\} <$><!--$$ , with $$\[--><$>a \in {\Bbb R} <$><!--$$ . The predicate P_a where $$\[--><$> P \in \{ \lt, \leq, =, \ne, \geq, \gt \} <$><!--$$ is interpreted as $$\[--><$> P_{a}^{{\cal D}} = \{ y \in {{\rDelta }^{\cal D}} \,|\,y\,P\,a\} <$><!--$$ . For example, $$\[--><$> \geq _{3}^{{\ \cal D}} = [3;\, + \,\infty [ <$><!--$$ . The DL constructors are extended for integrating concrete domains. It becomes then possible to define a concept $$\[--><$>\exists \tf="courier" g.\varphi <$><!--$$ , where g is a concrete role and $$\[--><$>\varphi \in {{\rPhi }^{\cal D}} <$><!--$$ . The concrete role g is interpreted as a partial function $$\[--><$>{{\tf="courier" g}^{\cal I}} <$><!--$$ : $$\[--><$>{{\rDelta }^{\cal I}} \mapsto {{\rDelta }^{\cal D}} <$><!--$$ where $$\[--><$>{{\rDelta }^{\cal I}} \, \cap \,{{\rDelta }^{\cal D}} = \emptyset<$><!--$$ and $$\[--><$>{{\left( {\exists {\tf="courier" g}.\varphi } \right)}^{\cal I}} = \left\{ {{\rm{\tf="courier" x}} \in {{\rDelta }^{\cal I}} |{{\tf="courier" g}^{\cal I}} \left( {\rm{\tf="courier" x}} \right) \in {{\varphi }^D} } \right\}. <$><!--$$

Accordingly, the concept of a ‘patient older than 70 having a tumor whose grade is different from 1’ can be represented as:

$$ {\rm{\tf="courier" TumorPatient}} \sqcap \exists {\rm{\tf="courier"hasAge}}{\rm{.}}{{{\rm{ \gt }}}_{{\rm{70}}}} \sqcap \exists {\rm{\tf="courier" hasTumor}}{\rm{.}}\left( {\exists {\rm{\tf="courier"hasGrade}}{\rm{.}}{{ \ne }_{\rm{\tf="courier" 1}}}} \right) $$

where TumorPatient is a concept representing patients suffering from a tumor.

2.3 Representation of a protocol using description logics

A protocol can be likened to a decision tree, where internal nodes are related to the patient characteristics and leaf nodes are recommendations of treatments (see Figure 1). An internal node corresponds to a binary or Yes/No question related to one or several patient characteristics or features. Examples of features are the age, the tumor size (size), the tumor grade (grade 1, 2, or 3), the status of the lymph nodes (N+ for involved and N− for non-involved lymph nodes), the status of the hormone receptors (HR+ for positive and HR− for negative hormone receptors). The protocol in Figure 1 shows that, for a patient without involved lymph nodes (N−), with positive hormone receptors (HR+), younger than 35 years, and with a tumor size >1 cm or having a grade g ≠ 1, a chemotherapy of level 1 is recommended.

Figure 1

Excerpts from the breast cancer treatment protocol used at the CAV Hospital in Nancy, France (www.oncolor.org)

A path in the tree corresponds to a conjunction of conditions satisfied by a patient. The set of patients for which all these conditions apply constitutes a class of patients. Such a path is represented by a concept in DL and is defined by characteristics checked at the nodes of the path. For example, the concept C1 represents the set of patients having no involved lymph nodes and positive hormone receptors:

$$\[--><$$l>no{ \qquad\quad{\rm{\tf="courier" C}}\tf="courier" 1 \equiv & {\rm{\tf="courier" TumorPatient}} \cr & \sqcap \exists {\rm{\tf="courier" hasLymphNodes}}{\rm{\tf="courier" .}}\left\{ {{\rm{\tf="courier" n - }}} \right\} \cr & \sqcap \exists {\rm{\tf="courier" hasHormoneReceptors}}{\rm{\tf="courier" .}}\left\{ {{\rm{\tf="courier" hr + }}} \right\} $$

The C1 concept reifies the path from the root of the tree to the node marked (1) in Figure 1. In particular, this concept includes the two nominals n− and hr+. Reasoning in presence of nominals is discussed in Section 4.1 and detailed for example in Lutz et al. (Reference Lutz, Areces, Horrocks and Sattler2005), Sirin et al. (Reference Sirin, Grau and Parsia2006), and Kazakov et al. (Reference Kazakov, Kroetzsch and Simancik2012). An instance PAT of the TumorPatient concept satisfying every condition on the path from the root of the tree to the node (1) is recognized as an instance of C1. The path leading to the edge marked by (2) in Figure 1 is represented by the concept C2:

$$\[--><$$l>no{ \qquad\quad{\rm{\tf="courier" C2}} \equiv {\rm{\tf="courier" C1}} \cr \ \qquad\sqcap \,\exists {\rm{\tf="courier" hasAge}}{\rm{\tf="courier" .}}{{{\rm{\tf="courier" \lt }}}_{{\rm{\tf="courier" 35}}}} \cr \qquad \ \sqcap \,\neg \left( {\exists {\rm{\tf="courier" hasTumor}}{\rm{\tf="courier" .}}\left( {\exists {\rm{\tf="courier" hasSize}}{\rm{\tf="courier" .}}{{{\rm{\tf="courier" \lt }}}_{\rm{\tf="courier" 1}}} \sqcap \exists {\rm{\tf="courier" hasGrade}}{\rm{\tf="courier" .}}\,{{{\rm{\tf="courier" = }}}_{\rm{\tf="courier" 1}}}} \right)} \right) $$

Following the physician way of doing, a cancer patient is considered as a person and as a medical case. A medical case is, in turn, considered as a problem case to be solved, or for which a solution has to be found, that is, a cancer patient has a tumor for which a treatment has to be recommended. This leads to the notion of medical case-treatment axiom (see [mct-ax]) in the Kasimir system (as made precise below, [mct-ax] is a kind of problem-solution axiom, see [ps-ax]). In the DL framework, the concepts to be considered are MedicalCase, TumorPatient, Tumor, and Treatment.

$$\[--><$$l> no{ \qquad\quad & {\rm{\tf="courier" TumorPatient}} \equiv {\rm{\tf="courier" Person}} \sqcap {\rm{\tf="courier" MedicalCase}} \sqcap \exists {\rm{\tf="courier" hasTumor}}{\rm{\tf="courier" .Tumor}} \cr & {\rm{\tf="courier" MedicalCase}} \sqsubseteq \exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .Treatment}} $$

A recommendation corresponds to a treatment to be applied: a specific treatment is represented by a subconcept of Treatment. For example, the [ax3] axiom represents the link existing between the class of patients C2 and a recommendation for a treatment based on a chemotherapy of level 1.

$$\[--><$$l>no {\qquad\quad{\rm{\tf="courier" C2}} \sqsubseteq \exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .Level1Chemotherapy}}$$

On a general level, the application of the protocol to a particular patient can be seen as a problem-solving task, where the problem corresponds to the description of the state of a cancer patient, that is, a medical case, and the solution to the description of a treatment that can be recommended according to the characteristics of the patient. The following elements are involved in the problem-solving model:

• • Problems are represented by the ProblemCase concept.

• • Solutions are represented by the SolutionCase concept.

• • Axioms relate specific problems to specific solutions.

In terms of DL representation, the concepts to be considered are:

$$\[--><$$l> no{\qquad\quad & {\rm{\tf="courier" ProblemCase}} \sqsubseteq \exists {\rm{\tf="courier" hasSolution}}{\rm{\tf="courier" .SolutionCase}} \cr & {\rm{\tf="courier" MedicalCase}} \sqsubseteq {\rm{\tf="courier" ProblemCase}} \cr & {\rm{\tf="courier" hasRecommendation}} \sqsubseteq {\rm{\tf="courier" hasSolution}} $$

The [ps-ax] axiom is called a problem-solution axiom and can be used to guide DL reasoning for solving domain-specific problems, for example, decision support in oncology. In addition, the MedicalCase concept is a subconcept of ProblemCase and the hasRecommendation role is a subrole of the hasSolution role. The introduction of problem-solution axioms is a generic technique for problem-solving, and can be reused in any other specific domain, for example, diagnosis, recognition, classification. In addition, problem-solution axioms provide a basis for integrating DLs and CBR as explained in Section 3.

2.4 Description logic reasoning for decision support

Given two concepts C and D in an ontology O, the subsumption test checks whether C is subsumed by D (or C is more specific than D): formally, C is subsumed by D, denoted by $$\[--><$>{\rm{\tf="courier" C}} \sqsubseteq {\rm{\tf="courier" D}} <$><!--$$ if, for every model $$\[--><$>{\cal I} <$><!--$$ of O, $$\[--><$>{{{\rm{\tf="courier" C}}}^{\cal I}} \sqsubseteq {{{\rm{\tf="courier" D}}}^{\cal I}} <$><!--$$ . Based on the subsumption test, the classification process consists in searching, for a given concept C, the concepts subsuming C and the concepts subsumed by C. The classification process allows to organize the concepts of an ontology into a concept hierarchy. In the Kasimir system, the concept hierarchy is used as a structure for representing a protocol.

The instance checking test checks whether an individual a is an instance of a concept C in an ontology O, that is, if for every model $$\[--><$>{\cal I} <$><!--$$ of O, $$\[--><$>{{{\rm{\tf="courier" a}}}^{\cal I}} \in {{{\rm{\tf="courier" C}}}^{\cal I}} <$><!--$$ . This test is a basis for the instantiation reasoning service that, given an individual a, consists in searching for the concepts C for which a is an instance of C. Decision support for protocol application is based on instantiation as follows. Relying on the problem-solution axiom [ps-ax], an individual recognized as an instance of the ProblemCase concept is linked to an instance of the SolutionCase concept through the hasSolution role.

Accordingly, an instance representing a cancer patient is first recognized as an instance of the concept MedicalCase and is then associated with an instance of the Treatment concept through the role hasRecommendation. For example, let us consider that the instance PAT is a cancer patient verifying the following assertions:

$$\[--><$$l>\qquad\quad \matrix{{{\rm{\tf="courier" TumorPatient}}\left( {{\rm{\tf="courier" PAT}}} \right)} \hfill \cr {{\rm{\tf="courier" Tumor}}\left( {\rm{\tf="courier" T}} \right)} \hfill \cr {{\rm{\tf="courier" hasTumor}}\left( {{\rm{\tf="courier" PAT ,}}\,{\rm{\tf="courier" T}}} \right)} {} \hfill \cr \\ {{\rm{\tf="courier" hasLymphNodes}}\left( {{\rm{\tf="courier" PAT,}}\,{\rm{\tf="courier" n - }}} \right)} \,\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \ \quad {\; \; \; \; no\,involved\,lymph\,node} \hfill \cr \\ {{\rm{\tf="courier" hasHormoneReceptors}}\left( {{\rm{\tf="courier" PAT,}}\,{\rm{\tf="courier" hr + }}} \right)} \,\qquad\qquad\qquad\qquad\qquad{\; \; \; \; positive\,hormone\,receptors} \hfill \cr \\ {{\rm{\tf="courier" hasAge}}\left( {{\rm{\tf="courier" PAT,}}\,{\rm{\tf="courier" 27}}} \right)} \qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \quad\quad\quad\quad\quad\quad\quad\quad\quad \ \ \quad {\; \; \; \; age\leq \it 35} \hfill \cr \\ {{\rm{\tf="courier" hasGrade}}\left( {{\rm{\tf="courier" T,}}\,{\rm{\tf="courier" 1}}} \right)} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \ {\; \; \; \; tumor\,of\,grade\,\it1} \hfill \cr \\ {{\rm{\tf="courier" hasSize}}\left( {{\rm{\tf="courier" T,}}\,{\rm{\tf="courier" 1}}{\rm{\tf="courier" .2}}} \right)} \quad \ \, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \ \ {\; \; \; \; tumor\,size\geq \it1} \hfill \cr \\ \\ \\ $$

The instantiation reasoning mechanism recognizes PAT as being an instance of the concept C2 (see axiom [ax2]). The axiom [ax3] states that a chemotherapy of level 1 is recommended for patients whose tumor satisfies the characteristics of the C2 concept. Formally, it can be inferred from the instantiation of C2 by PAT that there exists $$\[--><$> {\rm{\tf="courier" x}} \in {\rm{\tf="courier" Level1Chemotherap}}{{{\rm{\tf="courier" y}}}^{\cal I}} <$><!--$$ such that $$\[--><$> {\rm{\tf="courier" (PAT,}}\,{\rm{\tf="courier" x)}} \in {\rm{\tf="courier" hasRecommendatio}}{{{\rm{\tf="courier" n}}}^{\cal I}} <$><!--$$ , that is, the instance PAT is associated with the object x in $$\[--><$> {\rm{\tf="courier" Level1Chemotherap}}{{{\rm{\tf="courier" y}}}^{\cal I}} <$><!--$$ through the hasRecommendation role.

The concept satisfiability test checks whether a concept C can be instantiated, that is, if there exists a model $$\[--><$>{\cal I} <$><!--$$ of the ontology for which $$\[--><$>{{C}^{\cal I}} \ne \emptyset <$><!--$$ . In addition, an ontology O is said to be unsatisfiable or inconsistent if O does not have any model, for example, O includes an instance of an unsatisfiable concept. In the Kasimir system, the concept satisfiability test is used during the validation of the protocol representation, for example, for checking whether a decision rule can be triggered. Moreover, it is assumed that the description of a cancer patient is correctly defined with respect to the protocol, that is, the protocol and the description of the patient constitute a consistent knowledge base. The DL reasoner is then used during the design of the protocol and the elaboration of the patient description for detecting inconsistencies to be (manually) repaired.

3.1 Principles of case-based reasoning

CBR is used in many applications to solve problems in so-called ‘weak theory domains’. CBR solves new problems by retrieving and adapting solutions of previously solved cases stored in a case base (Riesbeck & Schank, Reference Riesbeck and Schank1989; Kolodner, Reference Kolodner1993; Aamodt & Plaza, Reference Aamodt and Plaza1994).

A typical CBR cycle is composed of four main steps: given a new problem, called a target problem, the retrieval step selects a source case from a case base; the adaptation step builds a solution to the target problem, relying on the differences between the source and target problems, and modifying accordingly the solution of the retrieved source case; the revision step improves the current target solution depending on actual and expected system results; finally the retention (memorization) step stores the newly solved case for further reuse. As shown in Figure 2, among these steps, retrieval and adaptation are closely related and have a central place.

Figure 2

The case-based inference. (1) A target problem is provided to the CBR system. (2) The system retrieves a source problem similar to the target problem. (3) The source problem is a problem associated with a solution. (4) The system adapts this solution in order to solve the target problem

In CBR, the basic knowledge unit is the case that represents a problem-solving episode. More precisely, a case (pb, Sol(pb)) is a pair composed of a problem pb and a solution Sol(pb) of pb. A case base is a structured set of source cases denoted by (srce, Sol(srce)). The CBR process aims at solving a target problem, denoted by tgt, with respect to a case base. During the retrieval step, a source problem srce, considered to be similar to the target problem tgt, is searched for in the case base. During the adaptation step, the solution Sol(srce) of srce is adapted for building a solution Sol(tgt) of tgt. Then, the solution Sol(tgt) may be tested and repaired. A learning step may complete the process where the new case (tgt, Sol(tgt)) can be memorized for future reuse.

In knowledge intensive CBR (KI-CBR) (Aamodt, Reference Aamodt1990, Reference Aamodt1991; Gómez-Albarrán et al., Reference Gómez-Albarrán, Gonzàles-Calero, Díaz-Agudo and Fernàndez-Conde1999; Aamodt, Reference Aamodt2004), the CBR process relies on a formalized model of domain knowledge, such as a domain ontology, for case organization and retrieval (Lieber & Napoli, Reference Lieber and Napoli1996). In Section 3.2, it is shown how a KI-CBR system takes advantage of domain knowledge for adapting a decision protocol, on the basis of the reformulation model.

3.2 Principles of case-based reasoning in the Kasimir system

The model of reformulations introduced in Melis et al. (Reference Melis, Lieber and Napoli1998) is a general framework for modeling and representing adaptation knowledge into simple and separated components. A reformulation is a pair $$\[--><$>({\rm{\tf="courier" r}},{{{\cal A}}_{\rm{\tf="courier" r}}}) <$><!--$$ where r is a relation between problems and $$\[--><$>{{{\cal A}}_{\rm{\tf="courier" r}}} <$><!--$$ is an adaptation function: when r relates srce to tgt, then any solution Sol(srce) of srce can be adapted into a solution Sol(tgt) of tgt, thanks to the adaptation function $$\[--><$>{{{\cal A}}_{\rm{\tf="courier" r}}} <$><!--$$ relating Sol(srce) to Sol(tgt) w.r.t. srce and tgt. Actually, $$\[--><$>{{{\cal A}}_{\rm{\tf="courier" r}}} <$><!--$$ is a function depending on three inputs, namely srce, Sol(srce), and tgt (this is made precise below).

The operations corresponding to problem relations r and adaptation functions $$\[--><$>{{{\cal A}}_{\rm{\tf="courier" r}}} <$><!--$$ have to be designed for a particular application, and, as well, a relevant strategy should be provided for building reformulations. Most of the time, and this is also true for the Kasimir system, reformulations are based on operations such as specialization, generalization, and substitution.

In the reformulation model, retrieval consists in finding a similarity path relating srce to tgt. Accordingly, a sequence of adaptation functions following the similarity path is reified into a corresponding adaptation path (see Figure 3).

Figure 3

On the first line, a similarity path from srce to tgt. On the second line, the corresponding adaptation path from Sol(srce) to Sol(tgt)

3.2.1 Similarity paths and adaptation paths

A similarity path SP from a problem srce to a problem tgt is a set SP of triples $$\[--><$>\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{{\rm{\tf="courier" i - 1}}}},\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" i}}},\,{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" i}}}} \right) <$><!--$$ , i ∈ {1, 2, …, q}, where pb₀ = srce and pb_q = tgt, r_i relates the pb_i−1 and pb_i problems and is such that there exists a reformulation $$\[--><$>({{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" i}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" i}}}}}) <$><!--$$ in the available adaptation knowledge (constituted by a finite set of reformulations). Given Sol(srce), a solution of srce, and a similarity path from srce to tgt, the solution of tgt can be built, thanks to a sequential application of the adaptation functions $$\[--><$>{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}}},\, \ldots, \,{\rm{ and}}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" q}}}}}<$><!--$$ :

$$\[--><$$l>{ \qquad\quad {\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 1}}}} \right)\, & = \,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_1}}}\left( {{\rm{\tf="courier" srce,}}\,{\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" srce}}} \right),\,{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 1}}}} \right) = {{{\cal A}}_{{{{\rm{\tf="courier" r}}}_1}}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 0}}},\,{\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 0}}}} \right),\,{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 1}}}} \right) \cr {\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 2}}}} \right)\, & = \,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_2}}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 1}}},\,{\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 1}}}} \right),\,{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" 2}}}} \right) \cr & \hskip -3.7pc\ldots \cr {\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" tgt}}} \right)\, & = \,{\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" q}}}} \right) = {{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" q}}}}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{{\rm{\tf="courier" q - 1}}}},\,{\rm{\tf="courier" Sol}}\left( {{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{{\rm{\tf="courier" q - 1}}}}} \right),\,{\rm{\tf="courier" p}}{{{\rm{\tf="courier" b}}}_{\rm{\tf="courier" q}}}} \right) \cr}$$

A similarity path is built according to a collection of available reformulations. Then several similarity paths relating two problems srce and tgt may exist. The choice between two similarity paths can be made either by the system according to an evaluation function or to an ‘external’ domain expert (as this is the case here). In the current implementation (see Section 5 and d'Aquin et al., Reference d'Aquin, Lieber and Napoli2006c for details), the set of available applicable operations, for example, specialization, generalization, substitution, is recursively used to enumerate the possible (non-cyclic) similarity paths. All possible solutions based on the different existing similarity paths are provided to the expert in charge of the decision.

This search for similarity paths respects the adaptation-guided retrieval principle (Smyth, Reference Smyth1996), where only source cases for which a solution is adaptable are retrieved, meaning and implying that adaptation knowledge is available. According to this principle, a similarity path provides a ‘symbolic reification’ of similarity between problems, allowing a case-based reasoner to build an understandable explanation of the results.

3.2.2 An example of application of reformulations to breast cancer treatment

Figure 4 presents a simplified example of the reformulation model for adapting the breast cancer treatment protocol. In this example, the tgt problem corresponds to the description of a patient for which the protocol cannot be directly applied. There are two reasons for that: (i) the patient is a man, and the protocol for breast cancer treatment takes into account only female patients, (ii) the tumor localization of this patient cannot be determined with precision in the breast, linked to the fact that the patient is a man. The other characteristics used for making a decision are related to the age of the patient, the size of the tumor, and the fact that the tumor has only one focal point.

Figure 4

An example of CBR problem-solving using reformulations, with similarity path from srce to tgt on first line and corresponding adaptation path from Sol(srce) to Sol(tgt) on the second line. CBR = case-based reasoning

Given the target problem tgt and the source case (srce, Sol(srce)), two reformulations, namely $$\[--><$>({{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}}) <$><!--$$ and $$\[--><$>({{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}}}) <$><!--$$ , are applied for solving tgt. Considering the similarity path, that is, reading the figure from right to left, from tgt to srce, relation r₂ denotes a substitution of Male Patient with Female Patient, while relation r₁ denotes a substitution of Unknown Localization with Internal Localization. Considering the adaptation path, the $$\[--><$>({{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}}) <$><!--$$ reformulation indicates that, when the localization of the tumor is unknown, the patient has to be considered as if she/he had an internal tumor, because it is the worst case, and thus, the one for which the largest treatment should be envisioned. The adaptation function $$\[--><$>{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}} <$><!--$$ simply corresponds to a copy of the solution Sol(srce). The $$\[--><$>({{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}},\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}}}) <$><!--$$ reformulation expresses that, since an ovary ablation is not applicable for a man, it has to be replaced with an hormone therapy having similar effects, that is, cures of an anti-oestrogen drug called tamoxifen.

For implementing a CBR system following this framework in a particular application domain, adaptation knowledge is needed, taking the form of a set of reformulations $$\[--><$>({\rm{\tf="courier" r}},\,{{{\cal A}}_{\rm{\tf="courier" r}}}) <$><!--$$ . This knowledge acquisition issue is addressed for the Kasimir system in two complementary ways: (1) the acquisition of adaptation knowledge from experts is examined in d'Aquin et al. (Reference d'Aquin, Lieber and Napoli2006b), that is, a kind of ‘manual’ knowledge acquisition process, whereas (2) a knowledge discovery approach is proposed in d'Aquin et al. (Reference d'Aquin, Badra, Lafrogne, Lieber, Napoli and Szathmary2007), that is, a kind of semi-automatic learning process based on frequent itemset search. The work presented here focuses on the reasoning mechanism making use of such adaptation knowledge and therefore assumes that it is available.

3.3 Description logic reasoning for supporting case-based reasoning inferences

The inference mechanisms available within DL formalisms, namely subsumption and instantiation, can be used for implementing a CBR system (Salotti & Ventos, Reference Salotti and Ventos1998; Gómez-Albarrán et al., Reference Gómez-Albarrán, Gonzàles-Calero, Díaz-Agudo and Fernàndez-Conde1999). The subsumption relation organizes the concepts of an ontology into a hierarchy, that can be used in CBR as an indexing structure for the case base, for facilitating the retrieval of relevant source cases (Lieber & Napoli, Reference Lieber and Napoli1996; Napoli et al., Reference Napoli, Lieber and Curien1996). In this hierarchy, an index idx(srce) is an abstraction of srce, encoding the relevant elements in the srce description leading to the Sol(srce) solution. The index idx(srce) is represented by the concept Idx_Srce and is linked to a solution represented by the Idx_Sol_Srce concept. A problem-solution axiom relates the index concept Idx_Srce to an index solution concept Idx_Sol_Srce as follows:

$$\[--><$$l>\qquad\quad {\rm{\tf="courier" Idx}}\_{\rm{\tf="courier" Srce}} \sqsubseteq \exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .Idx\_Sol\_Srce}} $$

Moreover, the source and target problems, respectively, denoted as srce and tgt in the CBR notation, are represented by instances in DLs, respectively, denoted by SRCE and TGT, while their solutions are represented by the instances SOL_SRCE and SOL_TGT. The individual SRCE is an instance of the index concept Idx_Srce, while the solution SOL_SRCE can be reused for any instance of the index concept Idx_Srce, that is, as soon as a problem TGT is recognized as an instance of Idx_Srce, the solution SOL_SRCE of SRCE can be reused to solve TGT.

The retrieval operation of CBR relies on instantiation for finding the index of a source problem having an adaptable solution. The similarity path built during retrieval relates the problem TGT to a problem PB₀ (i.e. the source problem), such that PB₀ is an instance of Idx_Srce:

$$\[--><$$l>\qquad\quad {\rm{\tf="courier" SRCE}} \, \hookrightarrow \,{\rm{\tf="courier" Idx\_Srce}}\, \hookleftarrow \,{\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 0}}}\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}\,{\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 1}}}\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}{\rm{\tf="courier" \ldots }}\,{\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" q}}}\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" q}}}\ {\rm{\tf="courier" TGT}} $$

The $$\[--><$>\hookrightarrow<$><!--$$ arrow stands for ‘is an instance of’. The source problem represented by an instance of the index concept Idx_srce is attached to a source solution through a problem-solution axiom similar to the axioms [mct-ax] and [ps-ax] introduced above. Thus, as soon as PB₀ is recognized as an instance of Idx_srce, the solution Sol_PB₀ is put in correspondence with the solution SOL_SRCE of SRCE. Therefore, the associated adaptation path is

$$\[--><$$l>\qquad\quad {\rm{\tf="courier" SOL\_SRCE}}\,{\rm{\tf="courier" = }}\,{\rm{\tf="courier" SOL\_P}}{{{\rm{\tf="courier" B}}}_0}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_1}}}\,{\rm{\tf="courier" SOL\_P}}{{{\rm{\tf="courier" B}}}_1}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_2}}} \ldots {\rm{\tf="courier" SOL\_P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" q}}}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" q}}}}}\,{\rm{\tf="courier" SOL\_TGT}} $$

As an application, let us consider the example introduced in Figure 4. The target problem (on the right) can be represented as

$$\[--><$$l>\qquad\quad \matrix{{{\rm{\tf="courier" TumorPatient(TGT)}}} \hfill \cr \\ {{\rm{\tf="courier" Male(S)}}} \hfill \cr \\ {{\rm{\tf="courier" hasGender(TGT,S)}}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \, \qquad {male\,gender}\hfill \cr \\ {{\rm{\tf="courier" hasAge(TGT,}}\,{\rm{\tf="courier" 55)}}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\qquad \qquad \quad \,\qquad{age\,between \ \it20 \ and \ \it 75}\hfill \cr \\ {{\rm{\tf="courier" hasTumor(TGT,}}\,{\rm{\tf="courier" T)}}} \hfill \cr \\ {\neg {\rm{\tf="courier" MultifocalTumor(T)}}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, \qquad unifocal \ tumor\hfill \cr \\ {{\rm{\tf="courier" UnknownLoc(L)}}} \hfill \cr \\ {{\rm{\tf="courier" hasLocalization(T,}}\,{\rm{\tf="courier" L)}}} \qquad \qquad \qquad \qquad \qquad \qquad \ \, \qquad {\; \; \; \; unknown\,tumor\,localization} \hfill \cr \\ {{\rm{\tf="courier" hasSize(T,}}\,{\rm{\tf="courier" 5}}{\rm{\tf="courier" .2)}}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \quad{\; \; tumor\,size\,between \ \it 4 \ and \ \it 7}\hfill $$

The index concept of the source problem in Figure 4 is represented by the concept Idx_Srce ([ax4]) with the problem-solution axiom introducing the concept Idx_Sol_Srce, that is, the recommended treatment ([ax45]):

$$\[--><$$l>no{ \qquad\quad {\rm{\tf="courier" Idx\_Srce}} \equiv & {\rm{\tf="courier" TumorPatient}} \cr & \sqcap \exists {\rm{\tf="courier" hasGender}}{\rm{\tf="courier" .Female}} \cr & \sqcap \exists {\rm{\tf="courier" hasAge}}{\rm{\tf="courier" .}}\,{{\geq }_{{\rm{\tf="courier" 20}}}} \cr & \sqcap \exists {\rm{\tf="courier" hasAge}}{\rm{\tf="courier" .}}\,{{\leq }_{{\rm{\tf="courier" 75}}}} \cr & \sqcap \exists {\rm{\tf="courier" hasTumor}}{\rm{\tf="courier" .}} \ {\rm{\tf="courier" (}}\neg {\rm{\tf="courier" MultifocalTumor}} \cr & \; \; \; \; \; \; \; \; \; \; \; \; \qquad \ \ \ \quad\sqcap \exists {\rm{\tf="courier" hasLocalization}}{\rm{\tf="courier" .Internal}} \cr \; \; \; \; \; \; \; \; \; \; \; \; & \qquad\qquad\qquad \ \quad\sqcap \exists {\rm{\tf="courier" hasSize}}{\rm{\tf="courier" .}}\,{{\geq }_{\rm{\tf="courier" 4}}} \cr \; \; \; \; \; \; \; \; \; \; \; \; & \qquad\qquad\qquad \ \quad\sqcap \exists {\rm{\tf="courier" hasSize}}{\rm{\tf="courier" .}}\,{{\leq }_{\rm{\tf="courier" 7}}})$$

$$\[--><$$l>no{\qquad\quad {\rm{\tf="courier" Idx\_Srce}} \sqsubseteq \exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .BreastAndOvaryAblation}} $$

The BreastAndOvaryAblation concept corresponds to Idx_Sol_Srce and represents treatments composed of a breast ablation and an ovary ablation. The similarity path built by the CBR retrieval operation corresponds to the similarity path shown in Figure 4:

$$\[--><$$l>\qquad\quad {\rm{\tf="courier" Idx\_Srce}}\,\hookleftarrow\,{\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 0}}}\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}\,{\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 1}}}\,{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}\,{\rm{\tf="courier" TGT}} $$

TGT is an instance of TumorPatient with the properties corresponding to the characteristics of the patient to be treated. PB₀ and PB₁ are also instances of TumorPatient. According to r₂, PB₁ is introduced with the same set of assertions as TGT, except for the gender:

$$\[--><$$l>\qquad\quad \matrix{ \matrix{{{\rm{\tf="courier" Female(S1)}}} \hfill \cr \\ {{\rm{\tf="courier" hasGender(P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 1}}},\,{\rm{\tf="courier" S1)}}} \qquad\quad \qquad\quad \qquad\quad \qquad\quad \qquad\quad \qquad\quad \qquad\quad\quad {\; \; \; \; female\,gender} $$

According to r₁, PB₀ (the source problem) is introduced with the same set of assertions as PB₁, except for the localization of the tumor:

$$\[--><$$l>\qquad\quad \matrix{{\neg {\rm{\tf="courier" MultifocalTumor}}({{{\rm{\tf="courier" T}}}_0})} \hfill \cr \\ {{\rm{\tf="courier" hasTumor}}({\rm{\tf="courier" P}}{{{\rm{\tf="courier" B}}}_0},\,{{{\rm{\tf="courier" T}}}_{\rm{\tf="courier" 0}}})} \hfill \cr \\ {{\rm{\tf="courier" Internal}}({{{\rm{\tf="courier" L}}}_0})} \hfill \cr \\ {{\rm{\tf="courier" hasLocalization}}({{{\rm{\tf="courier" T}}}_0},\,{{{\rm{\tf="courier" L}}}_{\rm{\tf="courier" 0}}})} $$

Then, PB₀ is recognized as an instance of the index concept Idx_Srce (see axiom [ax4]) and is linked to the solution SOL_PB₀ of the Idx_Sol_Srce treatment concept, that is, a surgery consisting of a breast ablation and an ovary ablation (see axiom [ax5]). The adaptation of the solution is based on the adaptation path:

$$\[--><$$l>\qquad\quad {\rm{\tf="courier" BreastAndOvaryAblation}}\,\hookleftarrow\,{\rm{\tf="courier" SOL\_P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 0}}}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}}\,{\rm{\tf="courier" SOL\_P}}{{{\rm{\tf="courier" B}}}_{\rm{\tf="courier" 1}}}\,{{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}}}\,{\rm{\tf="courier" SOL\_TGT}} $$

The instance SOL_PB₁ is a copy of SOL_PB₀ (application of $$\[--><$> {{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 1}}}}} <$><!--$$ ), and SOL_TGT is obtained by replacing the ovary ablation in SOL_PB₁ by cures of tamoxifen (application of $$\[--><$> {{{\cal A}}_{{{{\rm{\tf="courier" r}}}_{\rm{\tf="courier" 2}}}}} <$><!--$$ ):

$$\[--><$$l>\qquad\quad{\rm{\tf="courier" SOL\_TGT}}\, \hookrightarrow\,{\rm{\tf="courier" BreastAblationAndTamoxifen}}$$

4.1 The modularization of oncology protocols

Multiple viewpoints are visible in the structure of protocols. For example, the protocol for breast cancer treatment is based on several ‘sub-protocols’, each of them focusing on a particular phase of the medical treatment, that is, preoperative chemotherapy, surgery, radiotherapy, and complementary treatment, involving one or more specialties of oncology, among chemotherapy, surgery, radiotherapy, and hormonotherapy. A sub-protocol relies on its own vocabulary and provides a particular recommendation, in a given phase of the treatment, using only specific characteristics for this task. In addition, the breast therapeutic decision committee whose role is to adapt the content of the protocol for out-of-protocol cases, includes experts from different oncology specialties. An expert brings his/her own contribution to the solution, according to the specialty he/she depends on, and collaborates with the experts from other specialties, in the construction of a global and satisfactory solution.

DDLs are extensions of DLs in which several local ontologies are considered to be related through semantic mappings (Borgida & Serafini, Reference Borgida and Serafini2002, Reference Borgida and Serafini2003). Hereafter, it is shown how knowledge representation and reasoning based on the DDL framework is used within the Kasimir system for taking into account the multiple viewpoints of the oncology specialties. Actually, the representation of the protocol presented in Section 2.3 is modularized, meaning that it is reified as a multi-viewpoint representation relying on the principles of DDLs. Nevertheless, the characteristics of the representation introduced in Section 2.3 remain valid and are used locally in each viewpoint.

4.2 Distributed description logics and C-OWL

DDLs have been introduced in Borgida and Serafini (Reference Borgida and Serafini2002) and are the basis of the C-OWL extension of OWL for representing contextualized ontologies (Bouquet et al., Reference Bouquet, Giunchiglia, van Harmelen, Serafini and Stuckenschmidt2004). Contextualized ontologies are local representations of a domain, called contexts, that are semantically related through mappings. In the present framework, DDLs are used for formalizing and implementing several complementary representations of the oncology domain called viewpoints.

4.3 Representation of oncology viewpoints using distributed description logics

Modeling and formalizing a multiple viewpoint representation within a DDL is a decentralized task, that is, there is no need to set up a consensus, but there is a need to distinguish viewpoints. Three main steps can be considered (see Figure 5): (1) determine the relevant viewpoints in the domain, (2) build a local ontology for every viewpoint, and (3) establish mappings between local ontologies, reifying correspondences between viewpoints.

Figure 5

Three main steps can be considered for representing knowledge according to multiple viewpoints: determining viewpoints, building a local ontology, and mapping ontologies

4.3.2 Building a local ontology

Building a local ontology consists in formalizing a context in DDL corresponding to a viewpoint. A local ontology includes the knowledge units considered to be useful within the current viewpoint, independently from the other viewpoints, that is, a context in DDL formalizing a local ontology only implements knowledge units relevant to the viewpoint.

Decisions in the different specialties depend on different characteristics of the patient. Therefore, different concepts, representing a patient with different roles, are used for taking into account the protocol according to a given viewpoint. For example, in surgery, a patient with a directly operable and non-multifocal tumor has to be treated with a partial ablation of the breast. This is represented by the following axioms in O_s:

$$\[--><$$l>no{ \qquad\quad {\rm{\tf="courier" s}}{\rm{\tf="courier" :}}{\rm{\tf="courier" C}} \equiv & {\rm{\tf="courier" TumorPatient}} \cr & \sqcap \exists {\rm{\tf="courier" hasTumor}}{\rm{\tf="courier" .}}\, {\rm{\tf="courier" (OperableTumor}} \cr & \qquad\qquad \, \quad\qquad\sqcap \exists {\rm{\tf="courier" hasFocus}}{\rm{\tf="courier" .SimpleFocus)}} $$

$$\[--><$$l>no{\qquad\quad {\rm{\tf="courier" s}}{\rm{\tf="courier" :}}{\rm{\tf="courier" C}}\, \sqsubseteq \,\exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .PartialAblation}}$$

In radiotherapy, the considered characteristics are related to the recommended surgery, to the localization of the tumor in the breast, the size, and the presence of malignant cells in the lymph nodes:

$$\[--><$$l> no{\qquad\quad {\rm{\tf="courier" r}}:{\rm{\tf="courier" C}} \equiv & {\rm{\tf="courier" TumorPatient}} \cr & \sqcap \exists {\rm{\tf="courier" hasRecommendedSurgery}}{\rm{\tf="courier" .PreservingAblation}} \cr & \sqcap \exists {\rm{\tf="courier" hasLymphNodes}}{\rm{\tf="courier" .}}\left\{ {{\rm{\tf="courier" n - }}} \right\} \cr & \sqcap \exists {\rm{\tf="courier" hasTumor}}{\rm{\tf="courier" .}}\left( {\exists {\rm{\tf="courier" hasLocalization}}{\rm{\tf="courier" .InferoInternal}}} \right) $$

$$\[--><$$l> no{\qquad\quad {\rm{\tf="courier" r}}:{\rm{\tf="courier" RadiotherapyBreastAndEIMC}} \equiv & {\rm{\tf="courier" Irradiation}} \cr & \sqcap \exists {\rm{\tf="courier" hasZone}}{\rm{\tf="courier" .Breast}} \cr & \sqcap \exists {\rm{\tf="courier" hasZone}}{\rm{\tf="courier" .EIMC}} $$

$$no{\qquad\quad {\rm{\tf="courier" r}}{\rm{\tf="courier" :}}{\rm{\tf="courier" C}}\, \sqsubseteq \,\exists {\rm{\tf="courier" hasRecommendation}}{\rm{\tf="courier" .RadiotherapyBreastAndEIMC}} $$

These axioms state that a patient with a preserving surgery recommendation, non-involved lymph nodes, and a tumor located in the infero-internal part of the breast, has to be treated by a radiotherapy of the breast and a radiotherapy of the extended internal mammary chain (EIMC).

In the context O_ct the choice of the complementary treatment is guided by characteristics such as the status of the hormone receptors, the number of involved lymph nodes, the age of the patient, the size of the tumor, and the grade (1, 2, or 3). For example, the protocol states that a patient whose age is <35, with no involved lymph nodes, positive hormone receptors, a tumor of grade ≥2, or of size ≥1 cm, has to be treated by a chemotherapy of level 1:

$$\[--&#x003E;&#x003C;$$l&#x003E;no{ \qquad\quad {\rm{\tf=&#x0022;courier&#x0022; ct}}:{\rm{\tf=&#x0022;courier&#x0022; C}} \equiv &amp; {\rm{\tf=&#x0022;courier&#x0022; TumorPatient}} \cr &amp; \sqcap \exists {\rm{\tf=&#x0022;courier&#x0022; hasAge}}{\rm{\tf=&#x0022;courier&#x0022; .}}\,{{\leq }_{{\rm{\tf=&#x0022;courier&#x0022; 35}}}} \cr &amp; \sqcap \exists {\rm{\tf=&#x0022;courier&#x0022; numberInvolvedLymphNodes}}{\rm{\tf=&#x0022;courier&#x0022; .}}\left\{ {\rm{\tf=&#x0022;courier&#x0022; 0}} \right\} \cr &amp; \sqcap \exists {\rm{\tf=&#x0022;courier&#x0022; hasHormoneReceptors}}{\rm{\tf=&#x0022;courier&#x0022; .}}\left\{ {{\rm{\tf=&#x0022;courier&#x0022; hr + }}} \right\} \cr &amp; \sqcap \neg \exists {\rm{\tf=&#x0022;courier&#x0022; hasTumor}}{\rm{\tf=&#x0022;courier&#x0022; .}}\left( {\exists {\rm{\tf=&#x0022;courier&#x0022; hasGrade}}{\rm{\tf=&#x0022;courier&#x0022; .}}\,{{{\rm{\tf=&#x0022;courier&#x0022; = }}}_{\rm{\tf=&#x0022;courier&#x0022; 1}}} \sqcap \exists {\rm{\tf=&#x0022;courier&#x0022; hasSize}}{\rm{\tf=&#x0022;courier&#x0022; .}}{{{\rm{\tf=&#x0022;courier&#x0022; \lt }}}_{\rm{\tf=&#x0022;courier&#x0022; 1}}}} \right) $$

$$\[--&#x003E;&#x003C;$$l&#x003E;no{\qquad\quad {\rm{\tf=&#x0022;courier&#x0022; ct}}:{\rm{\tf=&#x0022;courier&#x0022; C}}\, \sqsubseteq \,\exists {\rm{\tf=&#x0022;courier&#x0022; hasRecommendation}}{\rm{\tf=&#x0022;courier&#x0022; .Level1Chemotherapy}}$$

4.3.3 Mapping ontologies

The relations existing between viewpoints are materialized through mappings between local ontologies. An important question concerns the knowledge units that can be shared and reused from a given viewpoint in another viewpoint. Thus, it is important to make precise the correspondences between viewpoints, indicating how a given context may take advantage of knowledge units in another context.

In the current example, a local ontology represents the vision of one or two specialties of oncology, playing a role in a particular phase of the treatment. Surgery is the central phase of the treatment and takes place after presurgical chemotherapy: then, decisions in surgery follow decisions in presurgical chemotherapy accordingly. In the same way, radiotherapy and complementary treatment aim at complementing surgery (for eliminating remaining involved cells with radiotherapy, chemotherapy, or hormonotherapy). The decision in these two viewpoints depends on the knowledge units in the surgery viewpoint. Figure 6 summarizes the existing mappings between the viewpoints introduced above. An arrow from a context O_i to a context O_j indicates the existence of a non-empty mapping $$\[--&#x003E;&#x003C;$&#x003E;{{{\cal M}}_{ij}} &#x003C;$&#x003E;&#x003C;!--$$ .

Figure 6

The mappings existing between viewpoints (contexts) in the multiple viewpoint representation of the protocol for breast cancer treatment

In this framework, bridge rules are used either for sharing knowledge on patients between the viewpoints or for representing the influence of a decision in a particular viewpoint on the decision in another viewpoint. For example, decisions in radiotherapy and in complementary treatment both rely on the status of the lymph nodes—either involved or not—even if this information unit is represented differently in O_r and O_ct. The sharing of this characteristic of the patient between O_r and O_ct can be represented by the bridge rules [br1] and [br2].

$$\[--&#x003E;&#x003C;$$l&#x003E; no{\qquad\quad {\rm{\tf=&#x0022;courier&#x0022; r}}: &amp; \exists {\rm{\tf=&#x0022;courier&#x0022; hasLymphNodes}}.\left\{ {{\rm{\tf=&#x0022;courier&#x0022; n - }}} \right\} \cr &amp; \mathop{\longrightarrow}\limits^{ \sqsubseteq }{\rm{\tf=&#x0022;courier&#x0022; ct \!\!:}}\exists {\rm{\tf=&#x0022;courier&#x0022; numberInvolvedLymphNodes}}{\rm{\tf=&#x0022;courier&#x0022; .}}\left\{ {\rm{\tf=&#x0022;courier&#x0022; 0}} \right\} $$

$$\[--&#x003E;&#x003C;$$l&#x003E; no{\qquad\quad {\rm{\tf=&#x0022;courier&#x0022; r}}: &amp; \exists {\rm{\tf=&#x0022;courier&#x0022; hasLymphNodes}}.\left\{ {{\rm{\tf=&#x0022;courier&#x0022; n + }}} \right\} \cr &amp; \mathop{\longrightarrow}\limits^{ \sqsubseteq }{\rm{\tf=&#x0022;courier&#x0022; ct \!:}}\exists {\rm{\tf=&#x0022;courier&#x0022; numberInvolvedLymphNodes}}{\rm{\tf=&#x0022;courier&#x0022; .}} \ {{\geq }_{\rm{\tf=&#x0022;courier&#x0022; 1}}} $$

In sequence, a decision in radiotherapy depends on the recommended treatment in surgery, either preserving, that is, a partial ablation preserving the essential part of the breast, or not preserving. The influence of the treatment in surgery on the treatment in radiotherapy is reified in the following bridge rule, allowing to complete a decision for the patient in O_r with the recommendation decided in O_s (bridge rule [br3]):

$$\[--&#x003E;&#x003C;$$l&#x003E; no{ \qquad\quad {\rm{\tf=&#x0022;courier&#x0022; s}}:\exists &amp; {\rm{\tf=&#x0022;courier&#x0022; hasRecommendation}}{\rm{\tf=&#x0022;courier&#x0022; .PartialAblation}} \cr &amp; \mathop{\longrightarrow}\limits^{ \sqsubseteq }\,{\rm{\tf=&#x0022;courier&#x0022; r:}}\exists {\rm{\tf=&#x0022;courier&#x0022; hasRecommendedSurgery}}{\rm{\tf=&#x0022;courier&#x0022; .PreservingAblation}} $$

4.4 Distributed description logic reasoning for decision support in oncology

In the Kasimir system, decision support based on multiple viewpoints is performed by global reasoning services provided by DDLs, mainly global instantiation (see Section 4.2.2). This reasoning service is extended to a decentralized CBR mechanism for case adaptation.

4.4.1 Global instantiation for protocol application

Solving a problem in a multi-viewpoint framework with CBR is a decentralized process: inferences occur locally, in each viewpoint represented by a context in DDL, taking advantage of bridge rules for reusing the knowledge and inferences from the other viewpoints. This problem-solving approach is similar to a reasoning process in decentralized artificial intelligence, defined in Demazeau and Müller (Reference Demazeau and Müller1989) as concerning the activity of autonomous intelligent agents that coexist and collaborate with other agents, each agent having proper goals and proper knowledge.

Hereafter, a specific example shows how global instantiation is used for applying the protocol in a decentralized reasoning approach. In the example, the considered patient is 33 years old, is suffering from breast cancer, with a tumor having the following characteristics: non-multifocal tumor of grade 2, with a size of 1.8 cm, directly operable, localized in the infero-internal part of the breast, with positive hormone receptors, and no involved lymph node. The physician having to establish a treatment for this patient may consider the above characteristics according to one or several viewpoints, among the four available viewpoints, that is, O_pc, O_s, O_r, and O_ct. However, according to domain knowledge, the viewpoint ‘presurgical chemotherapy’ cannot be considered in the present case. Thus, the physician is interested in the recommendations of the protocol for the three other viewpoints. The patient is represented by three local instances of the TumorPatient concept, namely: s:P (surgery), r:P (radiotherapy), and ct:P (complementary treatment) (see Figure 7).

Figure 7

The assertions describing the patient P to be treated in the three considered contexts

Equivalence bridge rules make precise the fact that the different instances introduced in the viewpoints denote a same patient:

$$\[--&#x003E;&#x003C;$$l&#x003E; no{ \qquad\quad &amp; {\rm{\tf=&#x0022;courier&#x0022; s}}:{\rm{\tf=&#x0022;courier&#x0022; P}}\,\mathop{\longrightarrow}^{ \equiv }\,{\rm{\tf=&#x0022;courier&#x0022; r}}\,{\rm{\tf=&#x0022;courier&#x0022; :}}\,{\rm{\tf=&#x0022;courier&#x0022; P}} \cr &amp; {\rm{\tf=&#x0022;courier&#x0022; r}}:{\rm{\tf=&#x0022;courier&#x0022; P}}\,\mathop{\longrightarrow}^{ \equiv }\,{\rm{\tf=&#x0022;courier&#x0022; ct}}\,{\rm{\tf=&#x0022;courier&#x0022; :}}\,{\rm{\tf=&#x0022;courier&#x0022; P}} \cr &amp; {\rm{\tf=&#x0022;courier&#x0022; ct:P}}\,\mathop{\longrightarrow}^{ \equiv }\,{\rm{\tf=&#x0022;courier&#x0022; s}}\,{\rm{\tf=&#x0022;courier&#x0022; :}}\,{\rm{\tf=&#x0022;courier&#x0022; P}} \cr} $$

Based on the assertions introduced in O_s and on the axioms associated with the protocol in O_s, namely [axS1] and [axS2], it can be inferred that a partial ablation is recommended for the patient in surgery (knowing that a directly operable tumor is an operable tumor). Accordingly, the bridge rule [br3] is used to complete the information known on the patient in radiotherapy, r:P is recognized as an instance of $$\[--&#x003E;&#x003C;$&#x003E;{\rm{\tf=&#x0022;courier&#x0022; r}}:\exists {\rm{\tf=&#x0022;courier&#x0022; hasRecommendedSurgery}}{\rm{\tf=&#x0022;courier&#x0022; . \ PreservingAblation}} &#x003C;$&#x003E;&#x003C;!--$$ .

Relying on the axioms [axR1], [axR2], and [axR3], in O_r, a radiotherapy of the breast and of the EIMC is recommended (RadiotherapyBreastAndEIMC). Now, the bridge rule [br1] allows to reuse information in O_r concerning involved lymph nodes in O_ct. Then, ct : P is considered as an instance of ct : ∃numberInvolvedLymphNodes.{0}. The axioms [axCT1] and [axCT2] in O_ct allow to infer that a chemotherapy of level 1 is recommended for the patient ct:P. Finally, the global solution recommended by the protocol is composed of three instances of Treatment, that is, three complementary viewpoints on the solution:

$$\[--&#x003E;&#x003C;$$l&#x003E; { \qquad\quad &amp;{\rm{\tf=&#x0022;courier&#x0022; s}}:{\rm{\tf=&#x0022;courier&#x0022; PartialAblation}} \cr &amp; {\rm{\tf=&#x0022;courier&#x0022; r:RadiotherapyBreastAndEIMC}} \cr &amp; {\rm{\tf=&#x0022;courier&#x0022; ct:Level1Chemotherapy}} \cr} $$

This decentralized problem-solving approach provides a decentralized solution composed of elements of solution for every viewpoint that has been considered during the problem-solving process. These three treatments do not represent alternatives, but different recommendations made by the system for the different phases of treatment, built from local knowledge in each viewpoint and taking into account the influence of viewpoints on one another.