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A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies

Maureen Donnelly1, Thomas Bittner1,2, Cornelius Rosse3

Department of Philosophy,

New York State Center of Excellence in Bioinformatics and Life Sciences,

University at Buffalo, 135 Park Hall, Buffalo, NY 142601;

Department of Geography,

University at Buffalo, Wilkeson Quad, Buffalo, NY 142612;

Structural Informatics Group, Department of Biological Structure,

University of Washington, Seattle, WA, USA3.

Correspondence: Maureen Donnelly, Department of Philosophy, University at Buffalo, 135 Park Hall, Buffalo, NY, 14260.

maureen.donnelly@ifomis.uni-saarland.de; telephone: 716-645-2444; fax: 716-645-6139.

Abstract

Objective: The objective of this paper is to demonstrate how a formal spatial theory can be used as an important

tool for disambiguating the spatial information embodied in biomedical ontologies and for enhancing their auto-

matic reasoning capabilities.

Method and Materials: This paper presents a formal theory of parthood and location relations among individuals,

called Basic Inclusion Theory (BIT). Since biomedical ontologies are comprised of assertions about classes of

individuals (rather than assertions about individuals), we define parthood and location relations among classes in

the extended theory BIT+Cl (Basic Inclusion Theory for Classes). We then demonstrate the usefulness of this

formal theory for making the logical structure of spatial information more precise in two ontologies concerned

with human anatomy: the Foundational Model of Anatomy (FMA) and GALEN.

Results: We find that in both the FMA and GALEN, class-level spatial relations with different logical properties

are not always explicitly distinguished. As a result, the spatial information included in these biomedical ontolo-

gies is often ambiguous and the possibilities for implementing consistent automatic reasoning within or across

ontologies are limited.

Conclusion: Precise formal characterizations of all spatial relations assumed by a biomedical ontology are neces-

sary to ensure that the information embodied in the ontology can be fully and coherently utilized in a

computational environment. This paper can be seen as an important beginning step toward achieving this goal,

but much more work is along these lines is required.

Keywords:

Ontology, Knowledge Representation, Anatomy, Mereology, Spatial Reasoning

1. Introduction

Spatial reasoning is a central component of medical research and practice and must be incorporated into any

successful medical informatics program. The spatial concepts most often used in biology and medicine are not

the quantitative, point-based concepts of classical geometry, but rather qualitative relations among extended ob-

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quantitative, point-based concepts of classical geometry, but rather qualitative relations among extended objects

such as body parts. The purpose of this paper is to propose a formal basis for the kind of qualitative spatial rea-

soning that is found in biology and medicine. We focus in this paper only on the most basic qualitative spatial

relations—parthood and location relations. But the general approach taken here can be extended to also include

other, more complex, qualitative spatial relations which are important in biomedical reasoning such as adjacency,

connectedness, and continuity.

Spatial reasoning in biology and medicine concerns either individuals or classes of individuals. By an individual

(also called a particular or an instance), we mean a concrete entity which, at each moment of its existence, occu-

pies a unique spatial location. Individuals can be either material (my liver, your brain) or immaterial (the cavity of

my stomach), where material individuals are here understood as those individuals with a positive mass and im-

material individuals are those individuals with no mass. Individuals are distinguished from classes (also called

universals, kinds, or types) which may have, at each moment, multiple individual instances. Examples of classes

are Liver (the class whose instances are individual livers) and White Blood Cell (the class whose instances are

individual white blood cells). (Throughout this paper, we use italics and initial capitals for class names.) Al-

though with time classes may gain and lose instances (when, e.g., white blood cells are created or die), the class

itself does not change its identity. In the design of biomedical ontologies, a special challenge is presented by the

need for associating spatial relations with classes, since in reality such relations hold only among individuals (see

below).

In recent years, much work has been done on constructing formal theories that model reasoning about qualitative

spatial relations among individuals [1-4]. A mereology is a formal theory of parthood and of relations--such as

overlap (having a common part) and discreteness (having no common part)--defined in terms of parthood. Since

its relations apply directly to concrete individuals and require neither quantitative data nor mathematical abstrac-

tions (points, lines, etc), a mereology is a natural basis for qualitative spatial reasoning in medicine.

In Section 2 of this paper, we present an extended mereology, Basic Inclusion Theory (BIT), which includes lo-

cation relations in addition to the usual mereological relations. By location relations, we mean relations that de-

pend only on the locations of relevant individuals and not on whether they share parts. Though not incorporated

into most mereologies, the distinction between mereological relations and location relations is crucial for medi-

cine since human bodies include immaterial spaces (cavities and lumina) which have no material parts but which

may contain material structures or substances. For example, a parasite (a material entity) may be located in an

intestinal lumen (an immaterial space) but the parasite is not itself part of the lumen or of the intestines and does

not share parts with them. Similarly, a portion of blood (a material substance) currently located in the cavity of

my right ventricle (an immaterial space) is not part of the right ventricle or its cavity.

All mereologies, including BIT, apply directly only to individuals such as my stomach or the lumen of a particu-

lar patient’s small intestine. A more complicated form of qualitative spatial reasoning -- reasoning about relations

among classes of individuals -- is also common in medical contexts. In canonical anatomy, we find assertions

such as "the stomach is continuous with the esophagus", "the right ventricle is part of the heart" or "the brain is

contained in the cranial cavity". As is emphasized in [5], it is important to distinguish these sorts of assertions

from claims about relations among individuals (e.g. "patient X’s right ventricle is part of patient X’s heart" or

"my stomach is continuous with my esophagus").

Since they apply to multiple individuals, the class-level relations are defined formally in terms of relations among

individuals using universal quantification. For example, [5] uses universal quantification and a mereologically-

formalized parthood relation to define relations among classes corresponding to the use of "part of" in assertions

of canonical anatomy such as "the right ventricle is part of the heart". In Section 3 of this paper, we show how

the same strategy can be used to define class-level versions of any relations among individuals, including all rela-

tions of BIT. Here we develop an extension of BIT, called Basic Inclusion Theory for Classes (BIT+Cl), which

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formally characterizes mereological and location relations among classes. In Section 4, we examine the logical

properties of the defined class relations. We find that different versions of the class relations have significantly

different logical properties. We also see that several important logical properties of the individual relations do

not transfer automatically to the corresponding class relations. Thus, though a strong formal theory of relations

among individuals is a necessary foundation for a formal theory of relations among classes, it is important to also

investigate the distinct logical properties of the class relations and to determine how they behave with respect to

particular kinds of classes.

A formal analysis of relations among classes, such as that presented in BIT+Cl, is critical for the development

and alignment of biomedical ontologies including the Foundational Model of Anatomy (FMA) [6], GALEN [7,

8], and the Gene Ontology (GO) [9], as well as terminologies such as the Systemized Nomenclature of Medi-

cine-Clinical Terms (SNOMED-CT) [10] and the Unified Medical Language System (UMLS). These ontologies

and terminologies consist mainly of claims about relations among biological classes. For example, in the FMA,

we have assertions such as: Right Ventricle part_of Heart; Liver contained_in Abdominal Cavity. In GALEN,

we have: Left Heart Ventricle isDivisionOf Heart; Liver isContainedIn Abdominal Cavity. (Throughout this

paper, we use Arial font for the relations of specific ontologies.) By establishing links between their relation

terms and the relations of a formal theory, the developers of a biomedical ontology can ensure that all curators

use their relation terms consistently within the biomedical ontology and make the meanings of their relation

terms clear to outside ontologists. In particular, formal analyses of the relation terms in the FMA and GALEN

are needed to determine whether these ontologies attribute the same meanings to similar terms (e.g. the FMA’s

contained_in vs. GALEN’s isContainedIn). In addition, formal analyses of relation terms are required for

strong, consistent automated reasoning within the ontologies. In Section 5 of this paper, we use BIT+Cl to ana-

lyze and compare the most general of the parthood and containment relations in the FMA and GALEN. We

show how precise and consistent characterizations of these relations would improve the clarity of the information

embodied in these ontologies and lead to stronger automated reasoning capabilities.

Because we focus at the end of this paper on the FMA and GALEN, our discussion throughout the paper con-

centrates on examples from human anatomy. However, the formal theory developed here is very general and can

be used to for reasoning about other kinds of classes of spatially or spatio-temporally located individuals (e.g.

classes of chemical substances or classes of diseases). In a different context, BIT+Cl could be used to describe

sub-processes of diseases or components of chemical substances.

2 Mereological and Location Relations among Individuals

Several different mereologies have been proposed in recent literature, for example [1, 2, 4]. Mereologies have

been extended to include also location relations in [3, 11]. In this section, we present a version of the formal

theory of [3] and discuss how it can be used to model medical reasoning about individual human bodies and the

parts and occupants of those bodies. We present the basic axioms, definitions, and theorems in sections 2.1 and

2.2. We call the formal theory consisting of these axioms and definitions Basic Inclusion Theory (BIT).

Notice that mereological and location relations may hold between individuals at some times but not at other

times. For example, the sinus venosus was part of my heart at an earlier developmental stage but no longer ex-

ists. Fully formed organisms also gain and lose parts: blood cells that are part of my body today will not be part

of my body in twenty days. However, for reasons of simplicity, mereologies typically do not deal with time and

change. We will follow that procedure and treat mereological and location relations throughout this paper as

time-independent relations. The theory thus developed here describes, within a given time-frame, a static spatial

arrangement of individuals. An important project for further work is to incorporate time and change into our the-

ory. Some progress is being made in this direction [12, 13].

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2.1 Mereological Relations

In this section, we introduce the basic mereological relations, axioms, and theorems. The theory is formulated in

first-order predicate logic with identity.

Parthood (symbolized as “P”) is the relation that holds between two individuals, x and y, whenever x is part of

y. In the mereologies of [3, 4, 11], parthood is treated as a primitive relation. This means that, instead of being

defined, axioms fixing the logical properties of the parthood relation are built into the theory. The parthood rela-

tion must then be interpreted in applications in a way that conforms to these axioms. Axioms that are included in

nearly every mereology are:

(P1)1 Pxx (every object is part of itself)2

(P2) Pxy & Pyx → x = y (if x is part of y and y is part of x, then x and y are identical)

(P3) Pxy & Pyz → Pxz (if x is part of y and y is part of z, then x is part of z)

(P1) tells us that P is reflexive, (P2) tells us that P is antisymmetric, and (P3) tells us that P is transitive. Thus, P

is a partial ordering (a reflexive, antisymmetric, and transitive binary relation). Axioms (P1)-(P3) are not very

strong. They cannot distinguish the parthood relation from other partial orderings such as the less-than-or-equal-

to relation on the real numbers or the is-a-factor-of relation on the positive integers. For this reason, most

mereologies include additional axioms which further restrict the parthood relation [14]. We suggest a few addi-

tional axioms that seem appropriate for anatomical reasoning in Section 2.3.

Proper parthood and overlap are binary relations among individuals that are defined in terms of parthood.

Proper Parthood: x is a proper part of y, if x is any part of y other than y itself. Symbolically:

PPxy =: Pxy & x ≠ y.

For example, my hand is a proper part of my body. My body is a part of itself, but it is not a proper part of itself.

Overlap: x and y overlap, if there is some object, z, that is part of both x and y. Symbolically:

Oxy =: ∃z (Pzx & Pzy).

My bony pelvis and my vertebral column overlap: my sacrum and my coccyx are part of both.

Inverse Relations: Inverses of the relations above may be introduced. The inverse of a binary relation S is the

binary relation S-1 defined: S-1xy if and only if Syx. (Here, S can be any binary relation, including a relation

among classes such as those introduced in Section 3. However, we focus now only on binary relations among

individuals.) Thus, PP-1xy if and only if PPyx. For example, PP-1(my heart, my right ventricle) tells us that my

heart has my right ventricle as one of its proper parts.

Notice however that when S is a symmetric relation (i.e. for all x and y, Sxy if and only if Syx), S-1 is the same

relation as S. For example, the overlap relation is symmetric and, therefore, is its own inverse (O-1 = O).

Additional relations (and their inverses) can be easily introduced into a mereology, but will not be considered in

this paper. For example, we could say that two individuals are discrete when they do not overlap (e.g. my brain

and my cranial cavity are discrete) and that two individuals properly overlap when they overlap but neither is

part of the other (e.g. my bony pelvis and my vertebral column properly overlap).

Basic Mereological Theorems: Because BIT is formulated in first-order predicate logic, we can derive an infi-

nite number of additional formulae from the axioms and definitions of BIT. These additional formulae are the

1 Axioms specific to the parthood relation are labeled with a "P".

2 Throughout this paper, initial universal quantifiers are dropped unless they are needed for clarity.

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theorems of BIT. Most of the theorems of any theory are uninteresting reformulations of the axioms and defini-

tions. But some are important logical consequences of the axioms and definitions that may not be obvious.

Even BIT’s relatively weak mereological axioms yield interesting theorems. Theorems such as the following are

useful for distinguishing the different mereological relations and for deriving additional assertions from one or

more input assertions about the mereological relations holding between specific individuals.

(PT1)3 PPxy & PPyz → PPxz (proper parthood is transitive)

(PT2) PPxy → ~PPyx (proper parthood is asymmetric: if x is a proper part of y, then y is not a proper part of x)

(PT3) ~PPxx (proper parthood is irreflexive: nothing is a proper part of itself)

(PT4) Oxy → Oyx (overlap is symmetric: if x overlaps y then y overlaps x)

(PT5) Oxx (overlap is reflexive: everything overlaps itself)

(PT6) PPxy → Oxy (if x is a proper part of y, then x overlaps y)

(PT7) Oxy & Pyz → Oxz (if x overlaps y and y is part of z, then x overlaps z)

For example, (PT1) tells us that from:

patient x’s left ventricle is a proper part of patient x’s heart

and

patient x’s aortic vestibule is a proper part of patient x’s left ventricle

it follows that

patient x’s aortic vestibule is a proper part of patient x’s heart.

2.2 Location Relations

Basic Inclusion Theory needs to be further extended to include also location relations among individuals. We can

already say something about the relative location of two objects using mereological relations: if x is part of y,

then x is located in y in the sense that x’s location is included in y’s location. Also, if x and y overlap, then x and

y partially coincide in the sense that x’s location and y’s location overlap. The location relations enable us to, in

addition, describe the relative location of objects that may coincide wholly or partially without being part of one

another or overlapping. A parasite in the interior of a person’s intestine is located in the lumen of his intestines,

but it is not part of the lumen of his intestines. As another example, my esophagus partially coincides with my

mediastinal space, but does not overlap (i.e. share parts with) my mediastinal space.

Human bodies have not only material parts (livers, hearts, etc) but also immaterial parts such as passageways and

spaces (the lumen of an esophagus, the cavities of the ventricles of a heart, an abdominal cavity) through which

substances pass and in which anatomical structures are located. Since the material entities which are temporarily

or permanently located in these spaces and passageways never share parts with them, mereological relations are

not useful for describing the positions of material individuals relative to spaces and passageways. For these rea-

sons, anatomical reasoning requires location relations distinct from mereological relations [15-18].

In both [3] and [11], all location relations are introduced in terms of a region function, r, that maps each individ-

ual to the unique spatial region at which it is exactly located at the given moment. Spatial regions are here as-

sumed to be the parts of an independent background space in which all individuals are located. Because we are

abstracting from temporal change and, in particular, from movement, we treat r as a time-independent primitive

function. BIT’s axioms for the region function are as follows.

(L1)4 Pxy → Pr(x)r(y) (if x is part of y, then x’s region is part of y’s region)

(L2) r(r(x)) = r(x) (x’s spatial region is its own spatial region)

The location relations are defined using the region function and mereological relations.

3 Theorems which can be derived from just the mereological axioms of BIT are labeled with "PT".

4 Axioms specific to the region function are labeled with "L".

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Located In: x is located in y if x’s region is part of y’s region. Symbolically:

Loc-In(x, y) =: Pr(x)r(y).

For example, my brain is located in (but not part of) my cranial cavity. A parasite may be located in (but not part

of) a patient’s intestinal lumen.

Partial Coincidence: x and y partially coincide if x’s spatial region and y’s spatial region overlap. Symbolically:

PCoin(x, y) =: Or(x)r(y) .

For example, my esophagus partially coincides with my mediastinal space. Notice that here the stronger relation

Loc-In does not hold. My esophagus’ region is not part of the region of my mediastinal space since part of my

esophagus lies outside of my mediastinal space. As another example, a bolus of food that is just beginning to en-

ter my stomach cavity partially coincides with (but is not located in) my stomach cavity.

Inverse Relations: Inverses of the relations above may be introduced. For example, x stands in the Loc-In-1 to y

if and only if Loc-In(y,x). Thus, Loc-In-1(my cranial cavity, my brain) tells us that my brain is located in my cra-

nial cavity.

Figure 1 is a composite of different configurations of the individuals x and y which can be distinguished in BIT.

Below each component of the figure, we list: first, the strongest relation (or conjunction of relations and their

negations) which holds from x to y; second, the strongest relation (or conjunction of relations and their nega-

tions) which holds from y to x; and third, an example of two anatomical individuals that stand in these relations.5

A solid line separating x and y indicates that x and y do not share any parts. A dotted line separating x and y in-

dicates that x and y do share parts.

Figure 1: Basic Spatial Inclusion Theory (BIT) relations

5 Note that the shapes of the drawings are not intended to correspond to the shapes of the individuals used as examples.

x

y

x y

x

y

x y

x y

PPxy

PP-1yx

(e.g., x is my left ventricle

and y is my heart)

Oxy

Oyx

(e.g. x is my vertebral

column and y is my

bony pelvis)

Loc-In(x, y) & ∼Oxy

Loc-In-1(y, x) & ∼Oyx

(e.g. x is a bolus of food and y

is my stomach cavity)

PCoin(x, y) & ∼Oxy

PCoin(y, x) & ∼Oyx

(e.g. x is my esophagus and y is my

mediastinal space)

∼PCoin(x, y)

∼PCoin(y, x)

(e.g. x is my heart and y is my

liver)

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As with the mereological relations, additional location relations could be easily added to BIT, but will not be

considered in this paper. For example, we could say that two individuals are non-coincident if they do not par-

tially coincide (e.g. my heart and my liver are non-coincident).

Theorems Involving Location Relations: From the axioms and definitions of BIT, we can derive the following

theorems concerning the location relations.

(LT1)6 Loc-In(x, x) (the located in relation is reflexive: every individual is located in itself)

(LT2) Loc-In(x, y) & Loc-In(y, z) → Loc-In(x, z) (the located in relation is transitive: if x is located in y and y

is located in z, then x is located in z)

(LT3) Pxy → Loc-In(x, y) (if x is part of y, then x is located in y)

(LT4) PPxy → Loc-In(x, y) (if x is a proper part of y, then x is located in y)

(LT5) Loc-In(x, y) & PPyz → Loc-In(x, z) (if x is located in y and y is a proper part of z, then x is located in z)

(LT6) PPxy & Loc-In(y, z) → Loc-In(x, z) (if x is a proper part of y and y is located in z, then x is located in z)

(LT7) PCoin(x, x) (partial coincidence is reflexive)

(LT8) PCoin(x, y) → PCoin(y, x) (partial coincidence is symmetric)

(LT9) Oxy → PCoin(x, y) (if x and y overlap, then x and y partially coincide)

(LT10) Loc-In(x, y) → PCoin(x, y) (if x is located in y, then x partially coincides with y)

Using, for example, (LT5) we can derive:

patient x’s heart is located in patient x’s thoracic cavity

from

patient x’s heart is located in patient x’s middle mediastinal space

and

patient x’s middle mediastinal space is a proper part of patient x’s thoracic cavity.

2.3 Additional Axioms

BIT’s restrictions on the mereological and location relations are rather weak. In particular, they are significantly

weaker than those of the theories presented in [1-4, 11]. As pointed out in Section 2.1, axioms (P1) - (P3) can-

not distinguish the parthood relation from very different partial orderings, such as the less-than-or-equal-to rela-

tion on the real numbers. The logical properties of BIT’s other relations are also only loosely constrained.

The purpose of this subsection is to briefly give a few examples of axioms that might be added to BIT to further

restrict the interpretations of its relations. It is important for the developers of a biomedical ontology to attempt

to link their relational terms to the relations of a strong formal theory. Even if additional axioms, such as those

listed here, are too complex to be implemented in an automated reasoning system, they can serve as guides to the

curators of the ontology and more precisely convey the intended understanding of the relational terms to outside

ontologists.

We mention here only restrictions that can be placed directly on the mereological relations. These restrictions

would in turn affect the other relations since the other relations are all delimited in terms of the parthood rela-

tion. For further examples of possible additional axioms (including axioms that apply directly to location rela-

tions) see [3, 11, 14].

6 Theorems that are derived using the region function axioms are labeled with "LT".

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The following principle cannot be derived from the axioms and definitions of BIT, but embodies an important

intuitive assumption about the mereological structure of concrete individuals such as body parts.

(*P4)7 PPxy → ∃z(PPzy & ∼Ozx) (if x is a proper part of y, then there is some proper part z of y that does not

overlap x)

(*P4) tells us that if an individual y has a proper part x then, since x does not comprise all of y, there must be at

least one proper part z that makes up some of what there is to y besides x. For example, since my right ventricle

is a proper part of my heart, there must be at least one proper part of my heart that does not overlap my right

ventricle. In fact there are several proper parts of my heart that do not overlap my right ventricle: my left ventri-

cle, my right and left atriums, my mitral valve, my aortic valve, and so on.

If added to BIT, (*P4) would allow us to derive the following theorem which prohibits individuals from having

only one proper part.

(*T1) PPxy → ∃z(PPzy & z ≠ x) (if x is a proper part of y, then y has some proper part besides x)

The following stronger axiom can be added to BIT instead of (*P4):

(*P5) If x is a proper part of y, then y has proper parts x1, ...,xn such that none of x, x1, ..., xn overlap and y is the

sum of x, x1, ..., xn.8

(*P5) tells us, for example, that since the body of my stomach is a proper part of my stomach, my stomach must

have other proper parts, namely, the fundus of my stomach and the pylorus of my stomach, such that none of

these parts overlap and, taken together, the three parts add up to my whole stomach. (In this case, we can say

that the collection consisting of the body of my stomach, the fundus of my stomach, and the pylorus of my stom-

ach form a partition of my stomach. See [13] for a formal treatment of partitions.)

As a final example, BIT could be further strengthened by the addition of the following axiom.

(*P6) ∀y∃x PPxy (for every individual y there is some individual x such that x is a proper part of y)

(*P6) tells us that every individual has some proper part. For example, my heart has millions of cells as proper

parts. The cells have membranes, cytoplasm, and nuclei as proper parts. And so on.

3. Relations among Classes

The assertions of canonical anatomy such as

the right ventricle is part of the heart

or

the brain is contained in the cranial cavity

are not limited to specific individuals but rather apply to all instances (or all normal instances) of the related ana-

tomical classes. On one interpretation, the first assertion tells us roughly that any right ventricle is part of a heart

and any heart has a right ventricle as a part. The second assertion can be interpreted as saying roughly that any

brain is contained in a cranial cavity and any cranial cavity contains a brain. Thus, these general statements imply

that certain spatial relations hold among very many specific individuals.

7 The labels for all additional axioms and theorems begin with an asterisk (*). The reader should keep in mind that these axioms are not included

in BIT and these theorems cannot be derived from the axioms of BIT.

8 (*P5) can be approximated formally, but the necessary formula is long and tedious and requires more formal machinery than we have introduced

in this paper.

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The purpose of this section is to present a general procedure for extending a formal theory of spatial relations

among individuals, such as BIT, to also include relations among classes corresponding to those made use of in

the two assertions above.

3.1. The Instantiation Relation

Since spatial relations hold directly only among concrete individuals, "spatial" relations among classes, such as

those assumed in the assertions of canonical anatomy, must be defined in terms of spatial relations among the

individual instances of the classes. Thus to define parthood and location relations among classes, we require, in

addition to the relations of BIT, a relation that links a class to its individual instances. We use here the time-

independent instantiation relation, Inst, of [5]. For a time-dependent version of this relation, see [13].

Following [5], we adopt the convention of restricting the variables x, y, z to individuals and using the variables

A, B, C, D for classes. All quantification is restricted to either the sub-domain of individuals or the sub-domain

of classes. Restrictions on quantification are not stated explicitly but can be understood from conventions on

variable usage.

For simplicity, we assume throughout the remainder of this paper that all anatomical classes are restricted to hu-

man anatomy, although we do not usually explicitly mention this restriction. Thus, Heart is the class of all human

hearts, White Blood Cell is the class of all human white blood cells.

The binary relation Inst holds between an individual x and a class A if x is an instance of A. In this case, we write

Inst(x, A).

For example, Inst(my heart, Heart) and Inst(my cranial cavity, Cranial Cavity).

Axioms for the instantiation relation include the following.

(I1)9 ∃x Inst(x, A) (every class has some member)

(I2) ∃A Inst(x, A) (every individual is a member of some class)

The Is_a subsumption relation between classes plays a key structuring role in most biomedical ontologies. It can

be defined in terms of Inst as follows.

Is_a(A, B) =: ∀x( Inst(x, A) → Inst(x, B))

This definition tells us that Is_a(A, B) (A is subsumed by B) means: every instance of A is also an instance of B.

For example Is_a(White Blood Cell, Cell) and Is_a(Heart, Organ).

We can also use the Inst relation and the overlap relation (O) of BIT to define a property of classes which will

turn out to be useful in our discussion of the logical properties of class relations below (Section 4). We will say

that class A is discrete if and only if no two instances of A overlap one another. Symbolically:

Discrete(A) =: ∀x ∀y(Inst(x, A) & Inst(y, A) & x ≠ y → ∼Oxy)

Most familiar examples of anatomical classes are discrete classes. For example, Heart, Liver, Cranial Cavity,

and Cell are all discrete classes-- two distinct hearts do not overlap, two distinct livers do not overlap, and so on.

Examples of non-discrete classes include many general classes such as Anatomical Structure, Organ System, or

Subdivision of Skeletal System (my alimentary system and my respiratory system are overlapping organ systems;

my bony pelvis and my vertebral column are overlapping subdivisions of my skeletal system) and substance

classes such as Blood or Urine (the portion of blood that is currently in the right side of my heart overlaps the

portion of blood that is currently in my right ventricle).

Notice that if a class A is discrete, then so are all of its subclasses.

(IT1) Discrete(B) & Is_a(A, B) → Discrete(A)

9 Axioms for the instantiation relation are labeled with "I". Theorems are labeled with "IT".

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Thus, for example, since Cell is a discrete class, the subclasses of Cell (Epithelial Cell, Muscle Cell, Neural

Cell, and so on) are all discrete classes.

3.2 Spatial Relations between Classes

Let T be any formal theory whose domain is restricted to individuals. T can be, for example, BIT or any other

formal theory of spatial relations among individuals. (In particular, T can be an extension of BIT which includes

more relations or more axioms than BIT.) T+Cl is the formal theory whose domain includes all individuals in the

domain of T plus classes of those individuals. The axioms of T+Cl are the axioms of T plus axioms (I1)-(I2). For

example, the axioms of BIT+Cl are (P1)-(P3), (L1)-(L2), and (I1)-(I2).

Let R be any binary relation from T. R is then a relation on individuals -- for example, the parthood relation (P),

the overlap relation (O), the located in relation (Loc-In), or any of the other relations of BIT. In T+Cl, we can

use R and the instantiation relation to define the following three relations among classes. (See also [5, 13, 19]

where these distinctions are made for different versions of class parthood relations. [20] uses description logic

for distinguishing versions of class parthood relations.)

R1(A, B) =: ∀x ( Inst(x, A) → ∃y( Inst(y, B) & Rxy))

R2(A, B) =: ∀y ( Inst(y, B) → ∃x( Inst(x, A) & Rxy))

R12(A, B) =: R1(A, B) & R2(A, B)

R1 class relations place restrictions on all instances of the first argument. R1(A, B) tells us that something is true

of all A’s -- each A stands in the R relation to some B.

R2 class relations place restrictions on all instances of the second argument. R2(A, B) tells us that something is

true of all B’s -- for each B there is some A that stands in the R relation to it.

R12 class relations place restrictions on all instances of both arguments. R12(A, B) tells us that something is true

of all A’s and something else is true of all B’s-- each A stands in the R relation to some B and for each B there is

some A that stands in the R relation to it.

As an example, we consider how three such class-level relations are defined when R is the proper part relation

(PP).

PP1 is the relation that holds between class A and class B if and only if every instance of A is a proper part of

some instance of B. For example, every instance of Human Female Reproductive System is a proper part of

some instance of Human Being. Thus, PP1(Human Female Reproductive System, Human Being).

PP2 is the relation that holds between class A and class B if and only if every instance of B has some instance of

A as a proper part. For example, every instance of Heart has an instance of Cell as a proper part. Thus,

PP2(Cell, Heart). But notice that PP2(Human Female Reproductive System, Human Being) does NOT hold,

since not all human beings have female reproductive systems. Also notice that PP1(Cell, Heart) does NOT hold,

since not all cells are part of a heart.

PP12 is the relation that holds between class A and class B if and only if: i) every instance of A is a proper part of

some instance of B and ii) every instance of B has some instance of A as a proper part. For example, every in-

stance of Human Nervous System is a proper part of some instance of Human Being and every instance of Hu-

man Being has some instance of Human Nervous System as a proper part. Thus, PP12(Human Nervous System,

Human Being). By contrast, neither PP12(Human Female Reproductive System, Human Being) nor PP12(Cell,

Heart) hold.

A few examples of assertions using other relations defined on classes are the following:

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O12(Bony Pelvis, Vertebral Column) (every bony pelvis overlaps some vertebral column and every vertebral col-

umn overlaps some bony pelvis)

O1(Male Genital System, Urinary System) (every male genital system overlaps some urinary system)

O2(Genital System, Male Urinary System) (every male urinary system overlaps some genital system)

Loc-In12(Brain, Cranial Cavity) (every brain is located in some cranial cavity and some cranial cavity has a brain

located in it)

Loc-In2(Blood, Cavity of the Right Ventricle) (blood is located in every cavity of a right ventricle)

PCoin12(Esophagus, Mediastinal Space) (every esophagus partially coincides with some mediastinal space and

every mediastinal space partially coincides with some esophagus)

For the purposes of this paper, we assume that assertions such as the following hold:

PP1(Cell Nucleus, Cell) (every cell nucleus is a proper part of some cell)

PP12 (Thumb, Hand) (every thumb is a proper part of some hand and every hand has some thumb as a proper

part).

To be precise, not every cell nucleus is part of a cell -- a cell nucleus can be removed from a cell. But normally

cell nuclei are parts of cells.10 Similarly, not every thumb is part of a hand and not every hand has a thumb as a

part, but normally thumbs are proper parts of hands and hands have thumbs as proper parts. Canonical anatomy

is concerned with anatomically normal individuals and not with aberrant cases. In a full theory of anatomical

classes, we will need a variant of the Inst relation (the normal-instance-of relation) that can distinguish the nor-

mal from abnormal instances of a class. But such a relation involves complications which go beyond the scope of

this paper. We do not deal here with abnormal instances of anatomical classes. In other words, we assume that

the domain of our theory is restricted to anatomically normal individuals. This policy is consistent with the

treatment of anatomical classes in the FMA. It also fits the treatment of classes of normal body parts (subclasses

of Intrinsically Normal Body Structure) in GALEN.

Finally, we note briefly that other strengths of class relations can be defined in terms of binary spatial relations on

individuals using either universal or existential quantification. For example, a much stronger type of class relation

than R1, R2, or R12 would hold between classes A and B only when all A’s stand in relation R to all B’s. A

weaker type of class relation than R1, R2, or R12 would hold between classes A and B when some A’s stand in

relation R to some B’s. (See [19] for other possibilities.) We do not explore such varieties of class relations in

this paper because they are not useful for analyzing (in Section 5) the current state of parthood and location as-

sertions for canonical anatomy in the FMA and GALEN. But such class relations could be useful either in some

other context or for expanding the type of anatomical information currently in the FMA and GALEN.

4 Reasoning about Relations among Classes

The axioms and definitions of BIT fix the logical properties of the spatial relations among individuals introduced

in that theory. However, most biomedical ontologies deal with relations between anatomical classes and not with

relations between individuals. We are thus particularly interested in determining the logical properties of class

relations such as those introduced by the definition schemas of Subsections 3.2.

10 Notice, however, that there are some cells (red blood cells) that do not normally have nuclei. Thus, even if we limit our domain to normal indi-

viduals, PP2(Cell Nucleus, Cell) does not hold.

Page 12

In this section, we discuss the logical properties of the R1, R2, and R12 types of class relations. Section 4 is di-

vided into two parts. Subsection 4.1 considers how the logical properties of the class relations correspond to the

logical properties of the underlying relations among individuals. Subsection 4.2 focuses both on the interaction

between R1, R2, and R12 relations and on the interaction between each of these relations and the Is_a (class sub-

sumption) relation. Throughout the section, we keep the discussion as general as possible, giving results that ap-

ply to T+Cl where T is any underlying formal theory of relations among individuals. But we frequently focus on

BIT+Cl for specific examples and list theorems of BIT+Cl that are useful for our discussion of the FMA and

GALEN in Section 5.

4.1 Transferring Properties of Individual Relations to Class Relations

Let T be, as above, any formal theory of relations among individuals. We consider here which of the logical

properties of the relations in T are inherited by the defined class relations in T+Cl. For example, if the relation R

in T is a strict partial ordering -- irreflexive, asymmetric, and transitive (as is the relation PP in BIT) -- does it

follow that in T+Cl that R1, R2, and R12 are also strict partial orderings? The answer is: not necessarily. When R

is a strict partial ordering, then each of R1, R2, and R12 must be transitive, but the class relations need not be irre-

flexive or asymmetric. For example, in BIT+Cl we can prove that each of PP1, PP2, and PP12 is transitive, but we

cannot prove that any of these relations are irreflexive or asymmetric.

We will see that in BIT+Cl, the R1, R2, and R12 class relations lack several logical properties of their BIT coun-

terparts. But first we discuss important properties of the relations among individuals that are transferred to at

least some of the class relations.

4.1.1 Transitivity. Let R be any transitive relation on individuals in theory T. Then in T + Cl, each of R1, R2, and

R12 is also transitive. Thus, since P, PP, and Loc-In are transitive relations of BIT, the class relations P1, P2, P12,

PP1, PP2, PP12, Loc-In1, Loc-In2, and Loc-In12 are all transitive.

(ClT1-3)11 Pi(A, B) & Pi(B, C) → Pi(A, C)

(ClT4-6) PPi(A, B) & PPi(B, C) → PPi(A, C)

(ClT7-9) Loc-Ini(A, B) & Loc-Ini(B, C) → Loc-Ini(A, C)

For example, it follows logically from

PP2(Cell, Heart)

(every heart has some cell as a proper part)

and

PP2(Heart, Cardiovascular System)

(every cardiovascular system has some heart as a proper part)

that

PP2(Cell, Cardiovascular System)

(every cardiovascular system has some cell as a proper part).

Also, it follows logically from Loc-In12(Heart, Middle Mediastinal Space) and Loc-In12(Middle Mediastinal

Space, Thoracic Cavity) that Loc-In12(Heart, Thoracic Cavity).

i = 1, 2, 1212

i = 1, 2, 12

i = 1, 2, 12

11 Theorems specific to BIT+Cl are labeled with “ClT”. We in general list explicitly only those theorems of BIT+Cl which are useful for our dis-

cussion of the FMA and GALEN in Section 5.

12 To save pointless repetitions, we frequently condense into one line three distinct theorems which differ only in indexing of the class relations.

Thus, for example, this line is a condensed representation of the following three BIT+Cl theorems:

(ClT1) P1(A, B) & P1(B, C) → P1(A, C)

(ClT2) P2(A, B) & P2(B, C) → P2(A, C)

(ClT3) P12(A, B) & P12(B, C) → P12(A, C).

Page 13

But care must be taken not to mix R1 and R2 class relations together in transitivity reasoning. For example, from

PP1(Uterus, Pelvis)

(every uterus is a proper part of a pelvis)

and

PP2(Pelvis, Male Human Being)

(every male human being has a pelvis as a proper part)

we cannot infer either

PP1(Uterus, Male Human Being)

(every uterus is a proper part of a male human being)

or

PP2(Uterus, Male Human Being)

(every male human being has a uterus as a proper part).

In general, for transitive R, Ri(A, B) & Rj(B, C) → Rk(A, C) holds only when i = j = k.13 For this reason, it is

important for biomedical ontologies that use more than one of the relations R1, R2, R12 for a given R (for exam-

ple, both PP1 and PP2) to explicitly distinguish these relations.

4.1.2 Reflexivity. Let R be any reflexive relation on individuals in theory T. Then the class relations R1, R2, and

R12 of T + Cl must be reflexive on the sub-domain of classes. Thus, P1, P2, P12, O1, O2, O12, Loc-In1, Loc-In2,

Loc-In12, PCoin1, PCoin2, and PCoin12 are reflexive relations on classes in BIT+Cl. For example, for any class A,

P12(A, A) -- each instance of A is part of some instance of A (itself) and each instance of A has some instance of

A (itself) as a part.

4.1.3 Symmetry. Let R be any symmetric relation on individuals in T. Then R12 must also be symmetric. Thus,

the relations O12 and PCoin12 of BIT + Cl are symmetric. For example, from

O12(Bony Pelvis, Vertebral Column)

(every bony pelvis overlaps some vertebral column and every vertebral column overlaps some bony pelvis)

we can in BIT+Cl derive:

O12(Vertebral Column, Bony Pelvis)

(every vertebral column overlaps some bony pelvis and every bony pelvis overlaps some vertebral column).

But R1 and R2 need not be symmetric class relations even if R is a symmetric relation among individuals. In

BIT+Cl, we may have O1(A, B) but not O1(B, A); O2(A, B) but not O2(B, A); PCoin1(A, B) but not PCoin1(B,

A); and PCoin2(A, B) but not PCoin2(B, A). For example, O1(Hand, Nerve) (every hand overlaps some nerve)

does NOT imply O1(Nerve, Hand) (every nerve overlaps some hand). Also PCoin2(Anatomical Cavity, Esopha-

gus) (every esophagus partially coincides with some anatomical cavity) does NOT imply PCoin2(Esophagus,

Anatomical Cavity) (every anatomical cavity partially coincides with some esophagus).

However, we can prove that if R is symmetric, then the following equivalence holds:

R1(A, B) ↔ R2(B, A)

Thus, O1(Hand, Nerve) implies, not O1(Nerve, Hand), but O2(Nerve, Hand). PCoin2(Anatomical Space,

Esophagus) implies, not PCoin2(Esophagus, Anatomical Space), but PCoin1(Esophagus, Anatomical Space).

Once again, we see that it is important for biomedical ontologies to explicitly distinguish class relations of type

R1, R2, and R12.

4.1.4 Simple Implications. Certain simple implications involving relations among individuals hold also for their

class relation counterparts. For example, let R and S be binary relations of T. Suppose that T includes a theorem

stating that for any individuals x and y

Rxy → Sxy

13 But note, as will be discussed in Subsection 4.2, that the stronger R12 relation may replace a R1 or R2 relation in the antecedent of a conditional

in the form of Ri & Rj → Rk . Thus, for example, for any transitive relation R, R1(A, B) & R12(B, C) → R1(A, C).

Page 14

Then in T+Cl we can prove that, for any classes A and B, all of the following hold:

R1(A, B) → S1(A, B)

R2(A, B) → S2(A, B)

R12(A, B) → S12(A, B).

For example, since PPxy → Loc-In(x, y) in BIT (theorem (LT3), subsection 2.2), we have the following theo-

rems in BIT+Cl:

(ClT10-12) PPi(A, B) → Loc-Ini(A, B)

Similarly, when either Rxy & Syz → Rxz or Sxy & Ryz → Rxz are theorems of T, then the three class relation

counterparts of each of these formulae are theorems of T+Cl. For example, from theorems (LT5) and (LT6) of

BIT (Subsection 2.2), we can derive the following theorems in BIT+Cl:

(ClT13-15) Loc-Ini(A, B) & PPi(B, C) → Loc-Ini(A, C)

(ClT16-18) PPi(A, B) & Loc-Ini(B, C) → Loc-Ini(A, C)

Thus, it follows from Loc-In12(Heart, Middle Mediastinal Space) and PP12(Middle Mediastinal Space, Thoracic

Cavity), that Loc-In12(Heart, Thoracic Cavity).

i = 1, 2, 12

i = 1, 2, 12

i = 1, 2, 12

As with transitivity inferences, implications that involve mixes of different types of class relations will not in gen-

eral be derivable. For example, neither Loc-In1(A, B) & PP2(B, C) → Loc-In1(A, C) nor Loc-In1(A, B) & PP2(B,

C) → Loc-In2(A, C) are theorems of BIT+Cl. This matches our intuitions about anatomical reasoning. From

Loc-In1(Prostate, Pelvic Cavity)

(every prostate is located in some pelvic cavity)

and

P2(Pelvic Cavity, Female Pelvis)

(every female pelvis has a pelvic cavity as a part)

we can infer neither

Loc-In1(Prostate, Female Pelvis)

(every prostate is located in some female pelvis)

nor

Loc-In2(Prostate, Female Pelvis)

(every female pelvis has some prostate located in it).

Also, implications involving negation, existential quantification, or a switch in the variables’argument places need

not transfer from the relations among individuals to their class relation counterparts. For example, we have al-

ready seen that O1(A, B) → O1(B, A) and O2(A, B) → O2(B, A) are not theorems of BIT +Cl, although Oxy →

Oyx is a theorem of BIT. We will see below more examples of implications involving relations among individuals

that do not carry over to the class relations.

4.1.5 Inverses. Recall that for any binary relation R in theory T, the inverse of R is the relation R-1 such that for

any individuals x and y

R-1xy ↔ Ryx.

In T+Cl, (R-1)12 must be the inverse of R12. In other words, we can prove in T +Cl that for any classes A and B

(R-1)12(A, B) ↔ R12(B, A).

In BIT+Cl, we have the following theorems:

(ClT19) (PP-1)12(A, B) ↔ PP12(B, A)

(ClT20) (Loc-In-1)12(A, B) ↔ Loc-In12(B, A).

Thus, it follows from PP12(Right Ventricle, Heart) that (PP-1)12(Heart, Right Ventricle) and vice versa.

However, inverse equivalences are not preserved for the weaker R1 and R2 class relations. In T+Cl, the following

equivalences do NOT in general hold:

Page 15

(R-1)1(A, B) ↔ R1(B, A)

(R-1)2(A, B) ↔ R2(B, A).

Instead, the following equivalences are derivable in T+Cl:

(R-1)2(A, B) ↔ R1(B, A)

(R-1)1(A, B) ↔ R2(B, A).

Thus, in BIT+Cl, (PP-1)2 is the inverse of PP1, (Loc-In-1)2 is the inverse of Loc-In1, (PP-1)1 is the inverse of PP2,

and (Loc-In-1)1 is the inverse of Loc-In2.

(ClT21) (PP-1)2(A, B) ↔ PP1(B, A)

(ClT22) (PP-1)1(A, B) ↔ PP2(B, A)

(ClT23) (Loc-In-1)2(A, B) ↔ Loc-In1(B, A)

(ClT24) (Loc-In-1)1(A, B) ↔ Loc-In2(B, A).

For example, PP1(Cell Nucleus, Cell) (every cell nucleus is a proper part of some cell) is equivalent to (PP-

1)2(Cell, Cell Nucleus) (for every cell nucleus there is some cell which has it as a proper part). PP1(Cell Nucleus,

Cell) is NOT equivalent to (PP-1)1(Cell, Cell nucleus) (every cell has some cell nucleus as a proper part). Once

again, we see the importance of distinguishing between the R1, R2, and R12 types of class relations.

4.1.6 Logical Properties of Relations which do not necessarily Transfer to Class Relations

Many of the theorems of the theory T need not hold in T+Cl for the class counterparts of the relations among

individuals. We have already seen above several examples of this discrepancy between the logical properties of

relations among individuals and the logical properties of the R1 and R2 types of class relations. Table 1 gives ad-

ditional information about which properties transfer automatically to the class relations and which do not.

Among Individuals

R is...

Reflexive

Irreflexive

Symmetric

Asymmetric

Antisymmetric

Transitive

Table 1: Correlation between the logical properties of a relation R for individuals and the logical proper-

ties of the class relations R1, R2, and R12

Among Classes

R2 must also be...?

Yes

No

No

No

No

Yes

R1 must also be...?

Yes

No

No

No

No

Yes

R12 must also be...?

Yes

No

Yes

No

No

Yes

For example, in BIT+Cl, we cannot prove that the relations PP12, PP1, and PP2 are irreflexive or asymmetric. In

particular, the following two formulae are NOT theorems of BIT+Cl:

~PP12(A, A)

PP12(A, B) → ~PP12(B, A)

We also cannot prove in BIT+Cl that the relations P12, P1, and P2 are antisymmetric. In particular, the following

formula is NOT a theorem of BIT+Cl:

P12(A, B) & P12(B, A) → A = B.

4.1.7 Discrete Classes. Recall from Section 3.1 that a discrete class is a class A such that no two instances of A

overlap. Recall also that many typical anatomical classes (e.g. Heart, Liver, Cell) are discrete. When reasoning is

restricted to a sub-domain of discrete classes, more of the logical properties of the relations of BIT are preserved

in the class relations. We can prove in BIT+Cl that, if all classes in a sub-domain D are discrete, then

i) PP1, PP2, PP12, (PP-1)1, (PP-1)2, and (PP-1)12 are irreflexive and asymmetric on D;