**4.1 Definition of objective graphs**

An objective graph � consists of the five components (N(�), R(�), E(�), L(�), C (�)), where


We define these components by induction on the structure of the node labels, as follows.

**Case 1**. Assume that the following data are given:

a. concept *C*,

200 Semantics – Advances in Theories and Mathematical Models

2. **Relations of patients and states**: The relations are defined between the [patient] and all state concepts. For example, the following relation denotes the relationship between

〈subject (of a state)⟩⊆ [patient]×[state of disease]. Note that these relations also share the same name "subject (of an state)" and that all concepts of states have the attributes of starting time points and terminating time points. We

3. **Relations of time ordering**: The relations are defined between the concepts of events and the states. For example, the following relations denote the relationships between

〈more than <p> before〉 ⊆ [operation]×[operation], 〈less than <p> befor〉 ⊆ [operation]×[operation], 〈less than <p> after〉 ⊆ [operation]×[operation] and 〈more than <p> after〉 ⊆ [operation]×[operation]. Here, "<p>" denotes a parameter. For example, the relation 〈before more than <2 weeks>〉 consists of a pair <op*1*, op*2*> if op*1* and op*2* are performed and if op*1* is performed more than

4. **Belonging relations of events**: The relations are defined between concepts of events with no term and events with terms. For example, the following relation denotes the

〈belonging〉 ⊆ [operation]×[hospital stay]. The relation contains a pair (op, sty) of an event of an operation op and that of a hospital

In this subsection, we define a graph that represents a target of quantification based on the medical service ontology defined in the previous subsection. We call such a graph an "objective graph". An objective graph is defined as a finite and labelled directed graph with a root node. A node in an objective graph is labelled by an instance of a concept or a value of an attribute of a concept in MSO, while an edge in an objective graph is labelled by an

An objective graph � consists of the five components (N(�), R(�), E(�), L(�), C (�)), where

We define these components by induction on the structure of the node labels, as follows.

relations between operations and hospital stays that have operations.

omit the explanation of the relations between patients and other states.

patients and their states of diseases.

stay sty if op is performed in the duration of sty.

**4. Representation of objects of quality indicators** 

operations.

two weeks before op*2*.

instance of a property in MSO.

i. N(�) is a set of nodes, ii. R(�) is a root node, iii. E(�) is a set of edges,

v. C(�) is a concept.

**4.1 Definition of objective graphs** 

iv. L(�) is a label function on N(�)∪E(�), and

**Case 1**. Assume that the following data are given:


Then, we define an objective graph �, as follows.

i. N(�):={\**0*, …, \**n*},


L(�)(*fi*):=*Ai* for *i=1,…, n*,

v. C(�):=*C*.

Note that if n=0, then N(�) is the singleton set {\**0*} and E(�) is the empty set.

**Case 2**. Assume that the following data are given:


Then, we define an objective graph �, as follows.


Each *f i* is called a main edge of � and each *f i,jk* is called an optional edge of �.

## **4.2 Example of an objective graph**

We give an example of an objective graph. For example, let us consider the quality indicator "5-year stomach cancer survival rate". The definition of the quality indicator is the ratio of the number of 5-year surviving patients to all stomach cancer patients, where a "stomach cancer patient" is a patient who had a diagnosis whose result was stomach cancer, and a "5 year surviving patient" is a patient who had a diagnosis whose result was stomach cancer but who is alive 5 years after that medical examination. Thus, we will first express the set of 5-year surviving patients in Figure 6. To this end, we construct three objective graphs �0, �1, and �2, as follows.

(1) �*0* = ({\*}, \*, ∅ (the empty set), *L0*, [patient]), where *L0*(\*)=[patient].

Representation System for Quality Indicators by Ontology 203

**Case 1.** If � is an objective graph defined in Case 1 of the definition of objective graphs, then

**Case 2.** Let � be an objective graph defined in Case 2 of the definition of objective graphs.

(iii) E(�\*) ⊆ E(�), where, for all \**<sup>i</sup>* ∈N(�\*)\{\**0*}3, the main edge from \**i* to \**0* in E(�) is

For the objective graph � in Fig. 6, the objective graph �\* in Fig. 7 is a segment of �, which

An objective graph � can be regarded to be a concept denoted by C(�) and modified by other concepts and properties that are denoted by L(�). If each concept is identified with the set of instances of the concept, an objective graph can be identified with a subset of the set denoted by C(�) that is obtained from C(�) by restricting it by sets and functions denoted by L(��. �o �a�e the i�entification clear, we here �efine an interpretation of an objective 

**Case 1.** Let � be an objective graph defined in Case 1 of the definition of objective graphs. Then, [[�]]:={c∈C | *c.A1*=*a1* ⋀ …⋀ *c*.*An*=*an* },4 where *c.Ai* is the value of the attribute *Ai* on *c*.

2 For sets *X* and *Y* with *Y X* and for a function *f* on *X*, *f*|*Y* denotes the function of *Y* that is defined by

Y}.

graph �\* defined in the following properties is a segment of �.

(iv) L(�\*) = L(�)|N(�\*)∪E(�\*) (the restriction of L(�) to N(�\*)∪E(�\*)), 2

Then, graph �\* defined in the following properties is a segment of �.

(iv) L(�)(\**i*):=�\**i* for all \**i*∈N(�\*), where �\**i* is a segment of �*i*,

for all *f i* ∈E(�\*) and

**4.4 Example of a segment of an objective graph** 

expresses the set of stomach cancer patients.

**5. Interpretation of objective graphs** 

**5.1 Definition of the interpretations of objective graphs**  For an objective graph �, we define a set [[�]], as follows.

*f*|*Y*(*y*) := *f*(*y*) for all y*Y*. We often refer to *f*|*Y* as the restriction of *f* to *Y*.

3 For sets *X*, *Y* with *YX*, *X*\*Y* denotes the set {*xX*| *x*

L(�)(*f j,ik*):= *Ri,jk* for all *f j,ik* ∈E(�\*).

(i) N(�\*) ⊆ N(�), (ii) R(�\*) = R(�), (iii) E(�\*) ⊆E(�),

(v) C(�\*)= C(�).

(i) N(�\*) ⊆ N(�), (ii) R(�\*) = R(�),

contained in E(�\*).

L(�)(*f i*):=*Ri*

(v) C(�\*) = C(�).

Fig. 7. A segment �\* of �.

graph, as follows.

(2) �*1* = ({\**0*, \**1*}, \**1*, {*f1*:\**0*→\**1*}, *L1*, [diagnosis]), where *L1*(\**0*) = [diagnosis], *L1*(\**1*) = ⟪stomach cancer⟫, *L1*(*f1*) = ⟨result⟩ and [diagnosis] denotes an event concept, ⟪stomach cancer⟫ denotes an instance of the concept of diseases, and ⟨result⟩ denotes an attribute of the concept [diagnosis]. Note that the range of ⟨result⟩ is the concept of diseases.

(3) �2 = ({\**0*, \**1*}, \**1*, {*f1*:\**0*→\**1*}, *L2*, [state of life or death]), where *L2*(\**0*) = [state of life or death], *L2*(\**1*) = ⟪true⟫, L2(f1) = ⟨survive⟩, [state of life or death] denotes the viability status of a patient, ⟪stomach cancer⟫ denotes an instance of the concept of diseases, and ⟨result⟩ denotes an attribute of the concept [diagnosis]. Note that the range of ⟨result⟩ is the concept of diseases.

We next construct an objective graph of "5-year surviving stomach cancer patients" G, as follows.

$$\begin{array}{l} \text{(i)} \ N(\mathbb{G}) = \{\ast\_{0} \, \*\_{1} \, \*\_{2}\}, \\ \text{(ii)} \ R(\mathbb{G}) = \ast\_{0} \\ \text{(iii)} \ E(\mathbb{G}) = \{f^{1} \colon \, \*\_{1} \to \, \*\_{0} \, f^{2} \colon \, \*\_{2} \to \, \*\_{0} \, f^{21} \colon \, \_{2} \xrightarrow{\ast} \, \_{1}\}, \\ \text{(iv)} \ L(\mathbb{G}) (\mathbb{A}\_{i}) = \mathbb{G}\_{i} \ (i = 0, 1, 2), \\ \qquad \qquad \qquad L(\mathbb{G}) (f^{1}) = \{\text{subject of the event}\} ) \ / \\ \text{L}(\mathbb{G}) (f^{2}) = \{\text{subject of the state}\} ), \\ \text{L}(\mathbb{G}) (f^{2}) = \{\text{after more than } \mathbb{G} \text{ years}\} ), \\ \text{(v)} \ \text{C}(\mathbb{G}) = \mathbb{C}(\mathbb{G}\_{0}) = \text{[patient]}. \end{array}$$

Fig. 6. Objective graph � ���������� 5-year surviving patients with stomach cancers

#### **4.3 Segments of an objective graph**

In the later section (Section 5), we will interpret an objective graph � as a set that is obtained from C(�) by adding the conditions defined by L(�). We define an objective graph �\*, which is called a segment of � and which can be interpreted as a super set of the interpretation of a given objective graph �, as follows.

**Case 1.** If � is an objective graph defined in Case 1 of the definition of objective graphs, then graph �\* defined in the following properties is a segment of �.

(i) N(�\*) ⊆ N(�), (ii) R(�\*) = R(�), (iii) E(�\*) ⊆E(�), (iv) L(�\*) = L(�)|N(�\*)∪E(�\*) (the restriction of L(�) to N(�\*)∪E(�\*)), 2 (v) C(�\*)= C(�).

**Case 2.** Let � be an objective graph defined in Case 2 of the definition of objective graphs. Then, graph �\* defined in the following properties is a segment of �.

(i) N(�\*) ⊆ N(�), (ii) R(�\*) = R(�), (iii) E(�\*) ⊆ E(�), where, for all \**<sup>i</sup>* ∈N(�\*)\{\**0*}3, the main edge from \**i* to \**0* in E(�) is contained in E(�\*).

(iv) L(�)(\**i*):=�\**i* for all \**i*∈N(�\*), where �\**i* is a segment of �*i*,

 L(�)(*f i*):=*Ri* for all *f i* ∈E(�\*) and L(�)(*f j,ik*):= *Ri,jk* for all *f j,ik* ∈E(�\*).

(v) C(�\*) = C(�).

202 Semantics – Advances in Theories and Mathematical Models

(2) �*1* = ({\**0*, \**1*}, \**1*, {*f1*:\**0*→\**1*}, *L1*, [diagnosis]), where *L1*(\**0*) = [diagnosis], *L1*(\**1*) = ⟪stomach cancer⟫, *L1*(*f1*) = ⟨result⟩ and [diagnosis] denotes an event concept, ⟪stomach cancer⟫ denotes an instance of the concept of diseases, and ⟨result⟩ denotes an attribute of the

(3) �2 = ({\**0*, \**1*}, \**1*, {*f1*:\**0*→\**1*}, *L2*, [state of life or death]), where *L2*(\**0*) = [state of life or death], *L2*(\**1*) = ⟪true⟫, L2(f1) = ⟨survive⟩, [state of life or death] denotes the viability status of a patient, ⟪stomach cancer⟫ denotes an instance of the concept of diseases, and ⟨result⟩ denotes an attribute of the concept [diagnosis]. Note that the range of ⟨result⟩ is the concept

We next construct an objective graph of "5-year surviving stomach cancer patients" G, as

Fig. 6. Objective graph � ���������� 5-year surviving patients with stomach cancers

In the later section (Section 5), we will interpret an objective graph � as a set that is obtained from C(�) by adding the conditions defined by L(�). We define an objective graph �\*, which is called a segment of � and which can be interpreted as a super set of the

concept [diagnosis]. Note that the range of ⟨result⟩ is the concept of diseases.

of diseases.

(1) �*0*=

(2) �*1*=

(3) �*2*=

follows.

(i) N(�) = {\**0*, \**1*, \**2*}, (ii) R(�) =\**0*,

(iv) L(�)(\**i*) = �*i* (*i=0, 1, 2*),

(v) C(�) = C(�*0*) = [patient].

(iii) E(�) = {*f <sup>1</sup>*:\**1*→\**0*, *f* 2:\**2*→\**0*, *f 21*:\**2*→\**1*},

 L(�)(*f 1*) = ⟨subject (of the event) ⟩ , L(�)(*f 2*) = ⟨subject (of the state) ⟩, L(�)( *f 21*) = ⟨after more than <5 years>⟩,

**4.3 Segments of an objective graph** 

interpretation of a given objective graph �, as follows.
