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.
firstname.lastname@example.org; telephone: 716-645-2444; fax: 716-645-6139.
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.
Ontology, Knowledge Representation, Anatomy, Mereology, Spatial Reasoning
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-
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
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 , 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,  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
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) , GALEN [7,
8], and the Gene Ontology (GO) , as well as terminologies such as the Systemized Nomenclature of Medi-
cine-Clinical Terms (SNOMED-CT)  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  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].
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 . 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.
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
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  and , 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".
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-
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.
(e.g., x is my left ventricle
and y is my heart)
(e.g. x is my vertebral
column and y is my
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
(e.g. x is my heart and y is my
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
patient x’s heart is located in patient x’s middle mediastinal space
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
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".
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
(*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  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
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.
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 . For a time-dependent version of this relation, see .
Following , 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
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
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".
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.  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
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,
A few examples of assertions using other relations defined on classes are the following:
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
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  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.
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
(every heart has some cell as a proper part)
PP2(Heart, Cardiovascular System)
(every cardiovascular system has some heart as a proper part)
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).
But care must be taken not to mix R1 and R2 class relations together in transitivity reasoning. For example, from
(every uterus is a proper part of a pelvis)
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)
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).
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)
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)
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:
(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.
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
R2 must also be...?
R1 must also be...?
R12 must also be...?
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, 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;
ii) P1, P2, P12, (P-1)1, (P-1)2, and (P-1)12 are antisymmetric.
Thus, for example, given that Heart and Right Ventricle are discrete classes, PP12(Right Ventricle, Heart) and
PP12 (Heart, Right Ventricle) cannot both hold. Also, for any discrete class A, none of the following can hold:
PP1(A, A), PP2(A, A), and PP12(A, A).
Of course, when A is a non-discrete class (e.g. Anatomical Entity or Blood), it may still be the case that none of
PP1(A, A), PP2(A, A), PP12(A, A) hold or that PP12(A, B) and PP12(B, A) do not both hold for any class B. But
these assertions cannot be derived in BIT+Cl.
4.1.8 Definitional Equivalences. In addition to the logical properties listed in TABLE 1, many of the equiva-
lences introduced in the definitions of BIT also do not carry over to the class relation setting and this is so even
when reasoning is restricted to a sub-domain of discrete classes. For example, according to the definition of the
overlap relation in BIT, for any x and y
Oxy ↔ ∃z(Pzx & Pzy)
(x and y overlap if and only if there is some individual z that is part of both x and y)
But none of the following equivalences is derivable in BIT + Cl:
O1(A, B) ↔ ∃C (P1(C, A) & P1(C, B))
O2(A, B) ↔ ∃C (P2(C, A) & P2(C, B))
O12(A, B) ↔ ∃C (P12(C, A) & P12(C, B))
The R1 and R2 versions of the equivalence are clearly not appropriate for anatomical reasoning. For example,
P1(Uterus, Pelvis) (every uterus is part of some pelvis) and P1(Uterus, Female Reproductive System) both hold,
but O1(Pelvis, Female Reproductive System) (every pelvis overlaps some female reproductive system) does
NOT hold. Also, P2(Cell, Heart) (all hearts have cells as parts) and P2(Cell, Liver) (all livers have cells as parts),
but NOT O2(Heart, Liver) (all livers overlap some heart).
However, the R12 version of the equivalence does seem plausible in an anatomical context. In fact, half of this
equivalence is derivable in BIT+Cl. It is a theorem of BIT+Cl that:
(ClT25) ∃C (P12(C, A) & P12(C, B)) → O12(A, B) (if there is a class C that stands in the P12 relation to both A
and B, then every instance of A overlaps an instance of B and vice versa).
The full equivalence would tell us that, in addition, whenever instance of A overlaps an instance of B and vice
versa, there is a class C that stands in the P12 relation to both A and B.
4.1.9 Conclusions. We draw at least two important conclusions from the points made in this subsection. First, as
emphasized throughout, it is crucial for biomedical ontologists to explicitly distinguish between R1, R2, and R12
Second, the logical properties imposed on relations among individuals in a formal theory may not automatically
transfer to the class relations that are defined in terms of them. This is one reason why it is important to always
clearly distinguish the class-level relations from the individual-level relations. In some cases, it is appropriate that
the logical properties of the individual-level relations do not transfer to the class-level relations. For example, O1
does not behave as a symmetric relation on anatomical classes (e.g. O1(Hand, Nerve) but not O1(Nerve, Hand)),
so it is an advantage of BIT+Cl that it does not force O1 to be symmetric. In other cases, the ontologist may find
it desirable to add axioms placing stronger restrictions directly on the class relations. For example, it seems plau-
sible that no anatomical class A (even a non-discrete class, such as Anatomical Entity or Blood) is such that
every instance x of A is a proper part of another instance of y of A.14 If so, an axiom stating that for any ana-
tomical class A, ~PP1(A, A) could be added to BIT+Cl. As another example, it seems plausible that whenever all
instances of A overlap instances of B and all instances of B overlap instances of A, there is some class C consist-
ing of those individuals which are the shared parts of A’s and B’s. If so an axiom stating
O12(A, B) → ∃C (P12(C, A) & P12(C, B))
could be added to BIT+Cl.
4.2 Reasoning about Relations among Classes: Additional Logical Properties of Class Relations
In this subsection, we present important logical properties of the defined class relations which do not correspond
directly to properties of the corresponding relations among individuals.
4.2.1 Logical Implications Involving Combinations of R12 and R1 or R12 and R2 Relations
One important property of the R12 class relations is that they always imply the corresponding R1 and R2 class re-
lations. More precisely, let T again be any formal theory of relations on individuals and let R be any binary rela-
tion in T. Then the following two implications hold:
R12(A, B) → R1(A, B)
R12(A, B) → R2(A, B)
For example, the following are theorems of BIT+Cl:
(ClT26-27) PP12(A, B) → PPi(A, B)
(ClT28-29) Loc-In12(A, B) → Loc-Ini(A, B)
(ClT26)-(ClT29) allow us to substitute the stronger R12 relations for the weaker R1 or R2 relations in the antece-
dent of another implication. For example, in combination with the transitivity theorems (ClT4) – (ClT9),
(ClT26)-(ClT29) yield the following additional theorems.
(ClT30-31) PPi(A, B) & PP12(B, C) → PPi(A, C)
(ClT32-33) PP12(A, B) & PPi(B, C) → PPi(A, C)
(CIT34-35) Loc-Ini(A, B) & Loc-In12(B, C) → Loc-Ini(A, C)
(ClT36-37) Loc-In12(A, B) & Loc-Ini(B, C) → Loc-Ini(A, C)
Thus, from PP1(Uterus, Pelvis) (every uterus is a proper part of some pelvis) and PP12(Pelvis, Trunk) (every
pelvis is a proper part of some trunk and every trunk has a pelvis as a proper part) , it follows that PP1(Uterus,
Trunk) (every uterus is a proper part of some trunk). As another example, from PP2(Cartilage, Vertebra) (every
vertebra has some cartilage as a proper part) and PP12(Vertebra, Vertebral Column) (every vertebra is a proper
part of some vertebral column and every vertebral column has some vertebra as a proper part) , it follows that
PP2(Cartilage, Vertebral Column) (every vertebral column has some cartilage as a proper part).
(ClT26) - (ClT29) also yield important further theorems when combined with theorems (ClT13)-(ClT18). Each
of the following can be derived in BIT+Cl:
(ClT38-39) PP12(A, B) & Loc-Ini(B, C) → Loc-Ini(A, C)
(ClT40-41) PPi(A, B) & Loc-In12(B, C) → Loc-Ini(A, C)
(ClT42-43) Loc-In12(A, B) & PPi(B, C) → Loc-Ini(A, C)
(ClT44-45) Loc-Ini(A, B) & PP12(B, C) → Loc-Ini(A, C)
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
i = 1, 2
14 But notice that the following axiom may not be desirable: for any anatomical class A, ~PP2(A, A). For example, it would seem that every in-
stance of Blood (i.e. any portion of blood) must have some instance of Blood (a smaller portion of blood) as a proper part.
For example, from PP12(Cervix of Uterus, Uterus) (every cervix of a uterus is a proper part of some uterus and
every uterus has a cervix of a uterus as a proper part) and Loc-In1(Uterus, Pelvic Cavity) (every uterus is lo-
cated in some pelvic cavity), it follows that Loc-In1(Cervix of Uterus, Pelvic Cavity) (every cervix of a uterus is
located in some pelvic cavity).
The BIT+Cl theorems listed above are important for our discussion in Section 5 of parthood and containment
relations in the FMA and GALEN. They are represented compactly along with theorems (ClT4) – (ClT9) and
(ClT13) – (ClT18) in TABLE 2.
PP1(B, C) PP2(B, C) PP12(B, C)
PP1(A, B) PP1(A, C) PP1(A, C)
PP2(A, C) PP2(A, C)
PP12(A, B) PP1(A, C) PP2(A, C) PP12(A, C)
Loc-In1(A, B) Loc-In1(A, C) Loc-In1(A, C)
Loc-In2(A, C) Loc-In2(A, C)
Loc-In12(A, B) Loc-In1(A, C) Loc-In2(A, C) Loc-In12(A, C)
TABLE 2: Inferences from conjunctions of PPi and Loc-Inj assertions
TABLE 2 tells us which relation between class A and class C can be inferred from a given assertion about the
relation between class A and class B (listed in row headings) in conjunction with an assertion about the relation
between class B and class C (listed in the column headings). For example, given PP2(A, B) (row 2) and Loc-
In12(B, C) (column 6), it follows from the axioms of BIT+Cl that Loc-In2(A, C) must also hold. (This is just
A blank cell in the table tells us that, unless additional information is given, we cannot derive any assertion of the
form Ri(A, B) where R is one of the relations of BIT. For example, from Loc-In1(A, B) (row 4) and PP2(B, C)
we cannot in general make any inference about the relation of class A and class C. To see this, consider the ex-
ample. Loc-In1(Prostate, Pelvic Cavity) (every prostate is located in a pelvic cavity) and PP2(Pelvic Cavity, Fe-
male Pelvis) (every female pelvis has a pelvic cavity as a proper part) both hold, but Ri( Prostate, Female Pelvis)
does not hold for any BIT relation R. In particular, Prostate stands in none of the relations PP1, PP2, PP12, Loc-
In1, Loc-In2, or Loc-In12 to Female Pelvis.
4.2.2 Logical Implications Involving R1, R2, R12 and Is_a
We can also derive many theorems describing the interaction between the R1, R2, and R12 relations and the Is_a
relation. In theory T +Cl where R is any binary relation among individuals in T, the following must hold for any
classes A, B, and C.
R1(A, B) & Is_a(B, C) → R1(A, C)
R1(A, B) & Is_a(C, A) → R1(C, B)
R2(A, B) & Is_a(A, C) → R2(C, B)
R2(A, B) & Is_a(C, B) → R2(A, C)
R12(A, B) & Is_a(A, C) → R2(C, B)
R12(A, B) & Is_a(C, A) → R1(C, B)
R12(A, B) & Is_a(B, C) → R1(A, C)
R12(A, B) & Is_a(C, B) → R2(A, C)
BIT+Cl theorems for the PPi relations corresponding to the schemata above are represented compactly in
Is_a(C, A) Is_a(A, C)
PP1(A, B) PP1(C, B)
PP2(A, B) PP2(C, B)
PP12(A, B) PP1(C, B) PP2(C, B)
TABLE 3: Inferences from conjunctions of PPi and Is_a assertions
The cells of TABLE 3 tell us i) which of the PPi relations must hold between A and C when a given PPi relation
holds between A and B (listed in the row headings) and a given subsumption relation holds between B and C
(listed in the column headings) and ii) which of the PPi relations must hold between C and B when a given PPi
relation holds between A and B (row headings) and a given subsumption relations holds between A and C (col-
umn headings). For example, given PP2(A, B) (row 2) and Is_a(C, B) (column 3), it follows that PP2(A, C) must
also hold. This corresponds to the BIT+Cl theorem:
PP2(A, B) & Is_a(C, B) → PP2(A, C).
A blank cell indicates that, unless further information is given, no inference of the form Ri(A, C), Ri(C, A), Ri(B,
C), or Ri(C, B) (with R a BIT relation) can be made. For example, from PP1(A, B) (row 1) and Is_a(C, B) (col-
umn 3), we cannot in general make any inference about the relation of A to C. To see consider the example:
PP1(Cell Nucleus, Cell) (every cell nucleus is a proper part of a cell) and Is_a(Platelet, Cell) (a platelet is a
cell), but no PPi relation holds between Cell nucleus and Platelet.
TABLE 4 is analogous to TABLE 3 but represents inferences involving the Loc-Ini relations rather than the PPi
TABLE 4: Inferences from conjunctions of Loc-Ini and Is_a assertions
For example, from Loc-In12(Ovary, Cavity of Female Pelvis) and Is_a(Cavity of Female Pelvis, Cavity of Pel-
vis), it follows (row 3/column 4) that Loc-In1(Ovary, Cavity of Pelvis). On the other hand, no assertion about
the Loc-Ini relation of Ovary to Cavity of Male Pelvis follows from Loc-In1(Ovary, Cavity of Pelvis) and
Is_a(Cavity of Male Pelvis, Cavity of Pelvis) (row 1/column 3).
Finally, we note that tables for the inverses of the PPi and Loc-Ini relations can be derived from TABLE 2 –
TABLE 4 and theorems (ClT21) – (ClT24) tying these relations to their inverses. For example, TABLE 5 repre-
sents BIT+Cl inferences from conjunctions of (Loc-In-1)i assertions and Is_a assertions.
Is_a(C, A) Is_a(A, C)
(Loc-In-1)1(B, A) (Loc-In-1)1(B, C)
(Loc-In-1)2(B, A) (Loc-In-1)2(B, C)
(Loc-In-1)12(B, A) (Loc-In-1)2(B, C) (Loc-In-1)1(B, C)
TABLE 5: Inferences from conjunctions of (Loc-In-1)i and Is_a assertions
5 Parthood and Containment Relations in the FMA and GALEN
In this section, we use the class relations introduced formally in BIT+Cl to analyze and compare class relations
used in the FMA and GALEN. We here select two biomedical ontologies with significant anatomical content and
focus on relations that roughly correspond to the PP1, PP2, PP12, Loc-In1, Loc-In2, Loc-In12 and their inverses in
The Foundational Model of Anatomy instantiates nearly 1 million part relations among its more than 70,000
classes. The FMA was developed over a ten year period by anatomists who, like the developers of most other
biomedical terminologies, were essentially unaware of spatial theories and of the requirements of formal knowl-
edge representation. Recent collaborations with theoreticians and knowledge engineers [5, 21], of which the cur-
rent communication is another example, provide opportunities for evaluating the FMA and for endowing it with
formal mechanisms that can enforce consistency and eliminate ambiguity.
The OpenGALEN Common Reference Model (CRM) was developed over a nine year period as a clinical ontol-
ogy resource. Like the FMA, GALEN’s CRM (and in particular the CRM’s anatomical component) was initially
constructed by domain experts with no prior training in knowledge representation. The subsequent development
of GALEN’s CRM, particularly the CRM’s high-level ontology, has benefited from theoretical work in ontology
and knowledge representation [7, 22].
OpenGALEN has a broader scope than the FMA, covering physiology, pharmacology, symptomatology, dis-
eases, and procedures in addition to human anatomy. The CRM anatomy sub-model, which includes most of
OpenGALEN’s assertions concerning class parthood and location relations, is approximately 25% the size of the
FMA. Unlike the FMA, the CRM anatomy sub-model deals with both normal and abnormal anatomy. However,
the level of detail of the anatomical information included in OpenGALEN is in general much coarser than that of
the FMA. In this paper, we consider only the CRM anatomy sub-model of OpenGALEN 6.
In collecting all data for this section, we used a version of the FMA dated from December, 2004 and the Open-
GALEN 6 Common Reference Model (Evaluation Edition) dated July 15, 2004.
5.1 Class Parthood in the FMA
The FMA has one general class parthood relation, part_of, which is divided into more specific sub-relations. For
example, the FMA distinguishes between anatomical parts and arbitrary parts. For this paper, we will not at-
tempt to distinguish these more specific class parthood relations. We focus exclusively on part_of and its in-
The FMA uses part_of as a proper parthood relation among anatomical classes, but does not (even with its
more specific parthod relations) explicitly distinguish between PP1, PP2, and PP12 uses of part_of. The FMA’s
part_of corresponds in different contexts to PP1, PP2, or PP12. For example, we find in the FMA:
the FMA’s part_of
Female Pelvis part_of Body
Male Pelvis part_of Body
Cavity of Female Pelvis part_of Abdominal Cavity
Urinary Bladder part_of Female Pelvis
Urinary Bladder part_of Male Pelvis
Cell part_of Tissue
Right Ventricle part_of Heart
Urinary Bladder part_of Body
Nervous System part_of Body
TABLE 6: Assertions using of the FMA’s part_of
Since, for example, every female pelvis is a proper part of some body but no male body has a female pelvis as a
part, part_of is used in 1a in the sense of PP1. On the other hand, since every female pelvis has a urinary bladder
as a proper part, but some urinary bladders (those belonging to men) are not part of any female pelvis, part_of is
used in 3a in the sense of PP2. The FMA uses part_of as the stronger relation PP12 only in examples such as 5 -
7, where every instance of the first class (e.g. Nervous System) is a proper part of some instance of the second
class (e.g. Body) and every instance of the second class has some instance of the first class as a proper part.
The FMA uses has_part as an inverse proper parthood relation among anatomical classes. No explicit distinc-
tions between (PP-1)1, (PP-1)2, and (PP-1)12 uses of has_part are made. However, inspection reveals that, for any
anatomical classes A and B, A has_part B is asserted in the FMA if and only if B part_of A is also asserted.
Thus, in contexts where part_of corresponds to PP1, has_part corresponds to the inverse of PP1, which is (PP-
1)2. In contexts where part_of corresponds to PP1, has_part corresponds to the inverse of PP2, which is (PP-1)1.
In contexts where part_of corresponds to PP12, has_part corresponds to the inverse of PP12, which is (PP-1)12.
Examples of the has_part relation in the FMA are:
the FMA’s has_part
Body has_part Female Pelvis
Body has_part Male Pelvis
Abdominal Cavity has_part Cavity of Female Pelvis
Female Pelvis has_part Urinary Bladder
Male Pelvis has_part Urinary Bladder
Tissue has_part Cell
Heart has_part Right Ventricle
Body has_part Urinary Bladder
Body has_part Nervous System
TABLE 7: Assertions using the FMA’s has_part
Transitivity of part_of in the FMA
The FMA allows unrestricted transitivity reasoning over its part_of relation. Thus in many cases, the FMA con-
A part_of C
from part_of assertions corresponding to
PPi(A, B) and PPj(B, C)
where i and j may be different indices. As can be easily seen from the upper left corner of TABLE 2 in subsection
4.2, the conjunction above supports an inference to a parthood assertion PPk(A, C) (for k = 1, 2, or 12) only
when either i) the indices i and j are identical or ii) at least one of i and j is the index 12.
In several cases, the FMA concludes A part_of C from a conjunction corresponding to PP2(A, B) and PP1(B,
C), even though we cannot in general infer from conjunctions of this form that one of the PPk relations holds be-
tween A and C. That the FMA does not by this procedure reach false conclusions is explained by special circum-
stances. In each of these cases, there is a fourth class D such that Is_a(B, D) (B is subsumed by D), PP12(A, D),
and PP12(D, C). Thus, the relations between D (the more general class) and each of A and C, guarantee that A
part_of C (here, in the sense PP12(A, C)).
The most common case of this type involves assertions about classes of sexually dimorphic structures. For ex-
ample, the unrestricted transitivity of part_of in the FMA allows us to derive
Urinary Bladder part_of Body
in the following ways:
Urinary Bladder part_of Female Pelvis & Female Pelvis part_of Body
Urinary Bladder part_of Male Pelvis & Male Pelvis part_of Body
If we make the distinctions between PP1, PP2, and PP12 explicit, we have
PP12 (Urinary Bladder, Body)
a*) from PP2(Urinary Bladder, Female Pelvis) & PP1 (Female Pelvis, Body)
b*) from PP2(Urinary Bladder, Male Pelvis) & PP1 (Male Pelvis, Body)
In this case, the following also hold: Is_ a(Female Pelvis, Pelvis), Is_a(Male Pelvis, Pelvis), PP12(Urinary Blad-
der, Pelvis) , and PP12(Pelvis, Body) (see Figure 2).15
Figure 2: The FMA’s part_of with Classes of Sexually Dimorphic Structures
Thus, the unrestricted transitivity of part_of yields in this case a true conclusion: Urinary Bladder part_of Body
(where this can be understood as: PP12(Urinary Bladder, Body)).
As far as we can see, the FMA’s unrestricted transitivity inferencing for part_of does not generate false asser-
tions. This is due partially to inherent features of human anatomy – e.g., that all human bodies are either male or
female. However, the FMA’s failure to distinguish the PP1, PP2, and PP12 meanings of the part_of relation
makes assertions using part_of ambiguous and leaves the logical structure of the knowledge embodied in the
Moreover, possibilities for expanding the FMA are limited unless distinctions between the different meanings of
part_of are made explicit. This expansion might include either i) additional explicit assertions about relations
between anatomical classes or ii) more sophisticated automated reasoning mechanisms. For i), there are many
useful assertions about parthood relations among anatomical classes which not only cannot be unambiguously
15 Note that Urinary Bladder part_of Pelvis and Pelvis part_of Body are not asserted in the FMA.
stated in terms of the part_of relation, but also would, if added to the FMA, lead to false conclusions. For ex-
ample, the FMA currently asserts Male Urethra part_of Urinary System. If the assertion Urinary System
part_of Female Pelvis (here in the sense PP2(Urinary System, Female Pelvis)) were added, the unrestricted
transitivity of part_of would yield the false conclusion: Male Urethra part_of Female Pelvis.
For ii), note that in all of the inference tables presented in subsection 4.2, the distinction between R1, R2, and R12
class relations is crucial for determining whether any inference can be made (and if so which one) from a con-
junction involving these relations. Thus, automated assertion generation based on these tables can be imple-
mented in the FMA only if PP1, PP2, and PP12 uses of part_of are explicitly distinguished.
We give a very simple example of how such automated reasoning might be advantageous for the FMA. The
FMA includes the assertions Ovary part_of Pelvis, Right Ovary subclass_of Ovary, and Left Ovary sub-
class_of Ovary (where subclass_of is the FMA’s Is_ a relation). No assertion is made about the relation of the
classes Right Ovary and Left Ovary to Pelvis. An automated reasoning mechanism based on TABLE 3 could
conclude both PP1(Right Ovary, Pelvis) and PP1( Left Ovary, Pelvis) from PP1(Ovary, Pelvis), Right Ovary
subclass_of Ovary, and Left Ovary subclass_of Ovary. On the other hand, no conclusion about parthood re-
lations between C and B follows from PP2(A, B) and Is_a(C, A). Thus, when we only have Ovary part_of Pelvis
without explicit information about which sense part_of is used in, we cannot automatically infer anything about
the relation of Right Ovary or Left Ovary to Pelvis.
In the subsection 5.5, we advocate that both the FMA and GALEN use distinguished versions of the relations
PP1, PP2, and PP12. There we sketch out further advantages of this approach.
5.2 Class Parthood in GALEN
According to the developers of GALEN, the GALEN version of a general class-level parthood relation is the
relation InversePartitiveAttribute . However, the logical properties of this relation are not clearly stipu-
lated in GALEN. In particular, InversePartitiveAttribute is not required to be transitive. We will therefore fo-
cus instead on the relation isDivisionOf which is the most extensively used of InversePartitiveAttribute’s two
immediate sub-relations. GALEN stipulates that isDivisionOf is transitive. It is distinguished from makesUp,
the other immediate sub-relation of InversePartitiveAttribute, by holding between classes of anatomical struc-
tures . By contrast, makesUp, but not isDivisionOf, may hold between classes of substances – e.g. Plasma
makesUp Blood. isDivisionOf is in this sense less general than the class-level proper parthood relations, PP1,
PP2, PP12, of BIT+Cl, since, e.g., PP12(Plasma, Blood). However, this particular discrepancy between the
BIT+Cl class parthood relations and isDivisionOf will not affect the discussion below since we consider only
examples involving classes of anatomical structures.
As with the FMA’s part_of, isDivisionOf has more specific sub-relations. These include: isSurfaceDivi-
sionOf, isSolidRegionOf, isLinearDivisionOf, isStructuralComponentOf, and isArbitraryComponentOf.
We will not attempt to distinguish between these sub-relations but will focus instead on isDivisionOf, their
An inspection of GALEN reveals that isDivisionOf is generally used as a restricted version of (i.e. a sub-relation
of) PP1. That is, in most contexts, if A isDivisionOf B is asserted in GALEN, then PP1(A, B) also holds – every
instance of A is a proper part of some instance of B. For example, GALEN asserts: Female Pelvic Cavity isDi-
visionOf Pelvic Part of Trunk, Prostate Gland isDivisionOf Genito-Urinary System, Prostate Gland isDivi-
sionOf Male Genito-Urinary System, and Left Heart Ventricle isDivisionOf Heart.
The GALEN relation hasDivision is generally used as a restricted version of (PP-1)1. That is, in most contexts, if
A hasDivision B is asserted in GALEN, then (PP-1)1(A, B) holds – every instance of A has some instance of B
as a proper part. For example, Pelvic Part of Trunk hasDivision Hair and Male Genito-Urinary System has-
Division Prostate Gland.
Recall that (PP-1)1 is NOT the inverse of PP1. Rather, (PP-1)1 is the inverse of PP2. (See Subsection 4.1.5.)
GALEN’s hasDivision is, correspondingly, NOT the inverse of isDivisionOf. In many cases, A isDivisionOf
B is asserted in GALEN, but B hasDivision A is not asserted. For example, Genito-Urinary System hasDivi-
sion Prostate Gland and Pelvic Part of Trunk hasDivision Female Pelvic Cavity are not asserted. In other
cases (but less often), B hasDivision A is asserted and A isDivisionOf B is not asserted. For example, Hair
isDivisionOf Pelvic Part of Trunk is not asserted.
GALEN generally asserts both A isDivisionOf B and B hasDivison A when the stronger relation PP12 holds
between A and B. For example, both Prostate Gland isDivisionOf Male Genito-Urinary System and Male
Genito-Urinary System hasDivision Prostate Gland are asserted. But hasDivision seems to be less regularly
used in GALEN than isDivisionOf. Thus, in several cases in which PP12(A, B) holds only A isDivisionOf B is
asserted. For example, Urinary Bladder isDivisionOf Genito-Urinary System is asserted, but Genitio-Urinary
System hasDivision Urinary Bladder is not asserted.
Finally, in a few contexts, isDivisionOf and hasDivision are used in a way that does not correspond to any of
the BIT+Cl class relations. For example, GALEN asserts Pericardium isDivisionOf Heart, as well as Heart
hasDivision Pericardium. But of the three classes of membranes which are subclasses of Pericardium only one,
Visceral Serous Pericardium (also called “epicardium”) has instances which coincide partially with instances of
Heart. The other two classes, Parietal Serous Pericardium and Fibrous Pericardium, have no instances which
even partially coincide with instances of Heart. Thus, not only the PPi relations, but also the much weaker
PCoini relations, fail to hold between Pericardium and Heart.
TABLE 8 summarizes different uses of GALEN’s isDivisionOf and hasDivision.
GALEN’s isDivisionOf assertion
Female Pelvic Cavity isDivisionOf Pelvic
Part of Trunk
Prostate Gland isDivisionOf Genito-
LeftHeartVentricle isDivisionOf Heart
Prostate Gland isDivisionOf Male Genito-
Urinary Bladder isDivisionOf Genito-
Pericardium isDivisionOf Heart
TABLE 8: Assertions using GALEN’s isDivisionOf and hasDivision
Pelvic Part of Trunk hasDivision Hair
Heart hasDivision LeftHeartVentricle
Male Genito-Urinary System hasDivision Prostate
Heart hasDivision Pericardium
5.3 Class Containment in the FMA
The FMA uses the relation contained_in as a class-level location relation. This relation is restricted so that
A contained_in B
may hold only when A is a class of material individuals and B is a class of immaterial individuals. In the FMA’s
terms, A must be a subclass of Material Physical Anatomical Entity and B must be a subclass of Anatomical
Space. Subclasses of Material Physical Anatomical Entity can be subclasses of either Anatomical Structure (e.g.
Heart) or subclasses of Anatomical Substance (e.g. Blood). Examples of subclasses of Anatomical Space include
Pelvic Cavity, Cavity of Stomach, and Lumen of Esophagus.
Because material individuals are never parts of immaterial individuals, A contained_in B and A part_of B can-
not both hold in the FMA. The mutual exclusivity of the FMA’s contained_in and part_of relations contrasts
with the inclusivity of the BIT+Cl relations Loc-In1, Loc-In2, and Loc-In12. By theorems (ClT10) – (ClT12),
Loc-Ini(A, B) must also hold whenever PPi(A, B) holds. For example, both PP12( Right Ventricle, Heart) and
Loc-In12( Right Ventricle, Heart) hold in BIT+Cl, whereas only Right Ventricle part_of Heart holds in the
The relation contains is used throughout the FMA as the inverse of contained_in. Thus, A contains B can
hold only when A is a subclass of Anatomical Space and B is a subclass of Material Physical Anatomical Entity.
The FMA uses contained_in in different contexts as a sub-relation of Loc-In1, a sub-relation of Loc-In2 or a
sub-relation of Loc-In12. Examples of these different uses of contained_in are given in TABLE 9.
the FMA’s contained_in
Right Ovary contained_in Abdominopelvic Cavity
Urinary Bladder contained_in Cavity of Female Pelvis
Urinary Bladder contained_in Cavity of Male Pelvis
Blood contained_in Cavity of Cardiac Chamber
Urinary Bladder contained_in Pelvic Cavity
Uterus contained_in Cavity of Female Pelvis
Prostate contained_in Cavity of Male Pelvis
Heart contained_in Middle Mediastinal Space
Blood contained_in Lumen of Cardiovascular System
Bolus of Food contained_in Lumen of Esophagus
TABLE 9: Assertions using the FMA’s contained_in
In example 7, every heart is located in some middle mediastinal space and every middle mediastinal space has a
heart located in it. By contrast, (example 1) although every right ovary is located in some abdominopelvic cavity,
some abdominopelvic cavities (those belonging to males) do not contain a right ovary. Thus, only Loc-In1(Right
Ovary, Abdominopelvic Cavity) holds. In example 3, every cavity of a cardiac chamber contains some portion
of blood, but not every portion of blood is located (at a specific time) in the cavity of a cardiac chamber (some
blood is instead in the lumen of the blood vessels). Thus, only Loc-In2(Blood, Cavity of Cardiac Chamber),
We note briefly that in a few examples involving anatomical substances, contained_in is not used as a sub-
relation of any of the BIT+Cl relations. For example 9, it is not the case that either i) every bolus of food is lo-
cated in the lumen of some esophagus or ii) every lumen of an esophagus has (at a given time) as bolus of food
located in it. In other words, neither Loc-In1(Bolus of Food, Esophagus) nor Loc-In2(Bolus of Food, Esopha-
gus) (as well as the stronger assertion Loc-In12(Bolus of Food, Esophagus)) holds. The assertion A con-
tained_in B seems in this and similar cases to mean that i) every instance of A is at some time located in some
instance of B and ii) every instance of B at some time has an instance of B located in it. This is a much more
complicated class relation than any considered in this paper since it assumes a time-dependent location relation
among individuals and requires quantification over times.
As with the part_of relation, an explicit distinction between the different uses of contained_in is essential for
disambiguating the FMA’s assertions. Different relations hold between the anatomical classes in examples 1, 2a,
4, and 9 above, but these differences are not made explicit in the FMA’s assertions.
Also as with part_of, a clear distinction between the different uses of the contained_in relation is necessary for
implementing automated reasoning over containment assertions. Currently, the FMA has no automated reason-
ing for the contained_in and contains relations. Note that, although contained_in is transitive, transitivity
reasoning over contained_in does not generate additional assertions. This is because the argument restrictions
on contained_in do not allow classes A, B, and C such that
A contained_in B & B contained_in C.
Since B cannot be both a class of immaterial individuals (as the second argument of contained_in in the first
conjunct) and a class of material individuals (as the first argument of contained_in in the second conjunct), the
conjunction above cannot hold. Thus, the antecedent of the transitivity implication is never satisfied and we can-
not generate additional assertions from the transitivity of contained_in.
But other of the BIT+Cl theorems embodied in Table 2 and Table 4 would be useful for generating further asser-
tions, if the Loc-In1, Loc-In2, and Loc-In12 uses of contained_in as well as the PP1, PP2, and PP12 uses of
part_of were clearly distinguished. For example, the FMA asserts
Heart contained_in Middle Mediastinal Space
Middle Mediastinal Space part_of Thoracic Cavity.
Since Loc-In12(Heart, Middle Mediastinal Space) and PP12(Middle Mediastinal Space, Thoracic Cavity), we
can use Table 2 (Row 6, Column 3) to infer: Loc-In12(Heart, Thoracic Cavity). Since, in addition, Thoracic
Cavity is a subclass of Anatomical Space, Heart contained_in Thoracic Cavity (with contained_in used as a
sub-relation of Loc-In12) should hold as well, but is not currently asserted in the FMA. See Figure 3.
Figure 3: Potential for reasoning about parthood and containment in the FMA
As another example, the FMA includes the assertions Lung contained_in Thoracic Cavity and Right Lung
subclass Lung. Since, Loc-In1(Lung, Thoracic Cavity), we can use Table 4 to derive Loc-In1(Right Lung, Tho-
racic Cavity). As a subclass of Lung, Right Lung must also be a subclass of Material Physical Anatomical En-
tity. Thus, Right Lung contained_in Thoracic Cavity (with contained_in used as a sub-relation of Loc-In1)
should also hold, but is not currently asserted in the FMA. See Figure 4.
Figure 4: Potential for reasoning about class subsumption and containment in the FMA
In general, BIT+Cl theorems concerning class location relations can be used to generate additional containment
assertions in the FMA as long as i) R1, R2, and R12 relations are distinguished and ii) if necessary, an extra step is
taken to check that the arguments of the derived BIT+Cl location assertion satisfy the FMA’s restrictions on the
arguments of contained_in.
5.4 Class Containment in GALEN
GALEN’s most general location relation is isContainedIn. Like isDivisionOf, isContainedIn has several sub-
relations. isPartitivelyContainedIn and isNonPartitivelyContainedIn are its two immediate sub-relations,
which are, in turn, each divided into several sub-relations. For the most part, the distinctions between the differ-
ent sub-relations of isContainedIn are not relevant to our discussion and will be ignored. However, we will
briefly mention below the special use of isPartitivelyContainedIn, since this sub-relation highlights one impor-
tant distinction between GALEN’s and the FMA’s containment relations.
Table10 lists examples of GALEN assertions using isContainedIn and its counterpart Contains.
In most—but not all—contexts, isContainedIn is used as a restricted version of Loc-In1. For example, GALEN
asserts Ovarian Artery isContainedIn Pelvic Cavity, Uterus isContainedIn Pelvic Cavity, Uterus isContain-
edIn Female Pelvic Cavity, and Mediastinum isContainedIn Thoracic Space.
The relation Contains is used in most contexts as a restricted version of (Loc-In-1)1. For example, GALEN as-
serts Venous Blood Contains Haemoglobin, Male Pelvic Cavity Contains Urinary Bladder, Female Pelvic
Cavity Contains Uterus, and Thoracic Space Contains Mediastinum. However, we have also found one con-
text in which Contains is used instead as a restricted version of (Loc-In-1)2 — GALEN asserts Pelvic Cavity
Contains Ovarian Artery, Pelvic Cavity Contains Uterine Artery, and Pelvic Cavity Contains Vaginal Artery.
Just as (PP-1)1 is not the inverse of PP1, (Loc-In-1)1 is not the inverse of Loc-In1. (See Subsection 4.1.5.) Thus,
as with isDivisionOf and hasDivision, isContainedIn and Contains are not inverses. For example, GALEN
asserts Uterus isContainedIn Pelvic Cavity, but not Pelvic Cavity Contains Uterus. Also, GALEN asserts Ve-
nous Blood Contains Haemoglobin, but not Haemoglobin isContainedIn Venous Blood.
Typically— but again not always — both A isContainedIn B and B Contains A are asserted when the stronger
Loc-In12 relation holds between A and B. For example, GALEN asserts Uterus isContainedIn Female Pelvic
Cavity and Female Pelvic Cavity Contains Uterus, as well as Mediastinum isContainedIn Thoracic Space and
Thoracic Space Contains Mediastinum. But note that Ovarian Artery isContainedIn Pelvic Cavity and Pelvic
Cavity Contains Ovarian Artery are both asserted even though Loc-In12(Ovarian Artery, Pelvic Cavity) does
NOT hold (instead only Loc-In2(Ovarian Artery, Pelvic Cavity) holds).
As additional exceptions to the typical behavior of isContainedIn and Contains, GALEN includes a significant
number of assertions of the form A isContainedIn B or B Contains A where none of the BIT+Cl relations
holds between A and B. For example, GALEN asserts Lung isContainedIn Pleural Membrane (as well as
Pleural Membrane Contains Lung). But no lung stands in the relation Loc-In to any pleural membrane.
GALEN seems to use isContainedIn in this and similar cases to indicate that members of one anatomical class
are surrounded by or enclosed within members of another anatomical class. This type of spatial relation is much
more complex than those introduced in BIT+Cl since it is based not just in topological structure but also re-
quires, at a minimum, some mechanism for distinguishing convex and non-convex structures (since only non-
convex individuals can surround other individuals). A slightly different example is the GALEN assertion Tooth
isContainedIn Tooth Socket (and also Tooth Socket Contains Tooth). Note that the relation between a tooth
and its socket is significantly weaker than the relation between a lung and its pleural membrane – a tooth socket
only partially surrounds its tooth.
Finally, we have found a small group of erroneous assertions which seem to appear in GALEN as a result of im-
proper automated reasoning over the Contains relation. GALEN implements unrestricted transitivity reasoning
on both isContainedIn and Contains. GALEN also seems to implement reasoning, corresponding roughly to
inferences represented in Tables 4 and 5, over conjunctions of isContainedIn and SubclassOf assertions or
conjunctions of Contains and SubclassOf assertions. As we have already seen, these kinds of inferences can
lead to false conclusions if the R1, R2, and R12 versions of class relations are not explicitly distinguished.
A failure to tailor automated reasoning to the different properties of Loc-In1, Loc-In2, Loc-In12, and their sub-
relations seems to be the reason for erroneous GALEN assertions such as Male Pelvic Cavity Contains Ovar-
ian Artery, Male Pelvic Cavity Contains Uterine Artery, and Male Pelvic Cavity Contains Vaginal Artery.
The assertion Male Pelvic Cavity Contains Ovarian Artery seems to have been generated from the GALEN
Pelvic Cavity Contains Ovarian Artery
Male Pelvic Cavity SubclassOf Pelvic Cavity.
These assertions correspond to the following BIT+Cl assertions:
(Loc-In-1)2(Pelvic Cavity, Ovarian Artery)
(every ovarian artery is located in some pelvic cavity)
Male Pelvic Cavity Is_a Pelvic Cavity
(every male pelvic cavity is a pelvic cavity).
See Figure 5.
Male Pelvic Cavity
Figure 5: Reasoning about containment and subclass relations in GALEN
As can been seen from Table 5 (row 2, column 3), no conclusion about location relations between classes A and
C can be derived from the conjunction (Loc-In-1)2(B, A) & Is_a(C, B).Thus, the inference from Pelvic Cavity
Contains Ovarian Artery and Male Pelvic Cavity SubclassOf Pelvic Cavity to Male Pelvic Cavity Contains
Ovarian Artery is invalid.
Ovarian Artery isContainedIn Pelvic
Uterus isContainedIn Pelvic Cavity
Uterus isContainedIn Female Pelvic
Mediastinum isContainedIn Thoracic
Larynx isContainedIn Neck
Lung isContainedIn Pleural Mem-
Tooth isContainedIn Tooth Socket
Pelvic Cavity Contains Ovarian Artery
2 Loc-In1 none
Venous Blood Contains Haemoglobin
Male Pelvic Cavity Contains Urinary Blad-
Female Pelvic Cavity Contains Uterus
5 Loc-In12 (Loc-In-1)12
Thoracic Space Contains Mediastinum (Loc-In-1)12
Neck Contains Larynx
Pleural Membrane Contains Lung
Tooth Socket Contains Tooth
Male Pelvic Cavity Contains Ovarian Ar-
TABLE 10: Assertions using GALEN’s isContainedIn and Contains
Examples in Table 10 highlight some important distinctions between the FMA’s and GALEN’s containment rela-
Examples 6 and 7 show that GALEN’s class containment relation does not, like that of the FMA, exclude class
parthood relations. GALEN uses the relation isPartitivelyContainedIn as a sub-relation of both isContaine-
dIn and isDivisionOf. Analogously, PartitivelyContains is in GALEN a sub-relation of both Contains and
hasDivision. These stronger relations hold between the pairs of anatomical classes in examples 6 and 7. Thus,
GALEN asserts both Mediastinum isContainedIn Thoracic Space and Mediastinum isDivisionOf Thoracic
Space, as well as both Larynx isContainedIn Neck and Larynx isDivisionOf Neck. Also, GALEN asserts both
Thoracic Space Contains Mediastinum and Thoracic Space hasDivision Mediastinum, as well as Neck Con-
tains Larynx and Neck hasDivision Larynx.
But note that GALEN’s isDivisionOf is not a sub-relation of isContainedIn . We have seen that in BIT+Cl,
PPi(A, B) implies Loc-Ini(A, B). By contrast, GALEN often asserts that A isDivisionOf B without also assert-
ing A isContainedIn B.16 For example, GALEN asserts Urinary Bladder isDivisionOf Lower Urinary Tract
but not Urinary Bladder isContainedIn Lower Urinary Tract. Also GALEN asserts Left Side Of Heart isDivi-
16 Of course, in these cases a sub-relation of isDivisionOf other than isPartitivelyContainedIn is used.
sionOf Heart, but not Left Side Of Heart isContainedIn Heart. Similarly, Contains is not a sub-relation of
hasDivision. For example, Lower Urinary Tract hasDivision Urinary Bladder and Heart hasDivision Left
Side Of Heart are asserted but not Lower Urinary Tract Contains Urinary Bladder and Heart Contains Left
Side Of Heart. It is not clear, however, exactly what principle GALEN uses to distinguish cases of class
parthood which are also cases of class containment from cases of class parthood which are not cases of class
Examples 6 and 7 (as well as examples 8 and 9) also demonstrate that GALEN does not, as the FMA does, re-
strict the arguments of its containment relation so that the first must be a class of material individuals and the
second must be a class of immaterial individuals. In GALEN, Mediastinum (the first argument in example 6) is a
subclass of Body Space and Neck (the second argument in example 7) is a subclass of Muscle Tissue Structure.
In sum, we have seen that GALEN’s failure to distinguish between (Loc-In-1)1 and (Loc-In-1)2 uses of Contains
has led in one case to false assertions and invalid inference mechanisms. Also, the root spatial meaning of
GALEN’s general containment relation is unclear. In particular, it is not clear what exactly (in the spatial con-
figurations the relevant individuals) is supposed to distinguish the isDivisionOf and isContainedIn relations. In
addition, isContainedIn and its sub-relations are used in some cases, not as a class-level location relations (like
BIT+Cl’s Loc-Ini relations or the FMA’s contained_in), but rather as class-level surrounds relations. The spa-
tial relation between my urinary bladder and my pelvic cavity is very different from the spatial relation between a
tooth and its socket. My urinary bladder occupies part of my pelvic cavity, while my tooth is partially surrounded
by its socket. Yet the same sub-relation of isContainedIn -- isNonPartitivelyContainedIn – is used in the
GALEN assertions about the corresponding classes: Urinary Bladder isNonPartitivelyContainedIn Pelvic-
Cavity and Tooth isNonPartitivelyContainedIn Tooth Socket.
5.5 Using BIT+Cl to Improve Anatomical Representation and Reasoning in Biomedical Ontologies
We recommend that all biomedical ontologies link their spatial terms to the relations of a formal theory, such as
BIT+Cl or (for more complex relations) an extension of BIT+Cl. We particularly urge that relations which are
distinct in the formal theory be linked to different relational terms in the biomedical ontology. This will greatly
improve the clarity of the information contained in the biomedical ontology – the user will know, e.g., that the
class parthood relation holding between Male Pelvis and Body is different from the class parthood relation hold-
ing between Urinary Bladder and Male Pelvis. It will also allow for expanded automated assertion generation
through the consistent implementation of reasoning based on theorems of the formal theory. The automated gen-
eration of assertions will, in turn, decrease the need for manual input into the ontology and, if consistently im-
plemented, decrease the number of erroneous assertions mistakenly entered into the ontology.
We have focused in Sections 4 and 5 on the need to explicitly distinguish R1, R2, and R12 types of class relations.
We now briefly sketch how a biomedical ontology which clearly distinguishes the PP1, PP2, and PP12 relations
might operate. For such an ontology, assertions involving the PP1 and PP2 relations could be generated auto-
matically from PP12 assertions and Is_a assertions. In addition, the transitivity of PP12 can be used to automati-
cally generate PP12 assertions from a (relatively) small collection of manually entered PP12 assertions. For exam-
ple, given the following inputs:
PP12(Urinary Bladder, Pelvis)
PP12(Neck of Urinary Bladder, Urinary Bladder)
transitivity reasoning on PP12 generates:
PP12(Urinary Bladder, Body)
PP12(Neck of Urinary Bladder, Body)
PP12(Neck of Urinary Bladder, Pelvis).
Given also the Is_a assertions:
Is_a(Female Pelvis, Pelvis)
Is_a(Male Pelvis, Pelvis)
the theorems represented in Table 3, generate:
PP2(Urinary Bladder, Female Pelvis)
PP2(Urinary Bladder, Male Pelvis)
PP2(Neck of Urinary Bladder, Female Pelvis)
PP2(Neck of Urinary Bladder, Male Pelvis)
See Figure 2, Subsection 5.1.
Note that the strong PP12 relations can link to Pelvis or Body only classes, like Urinary Bladder, whose in-
stances are parts of all pelvises and bodies. But the PP12 relation can link classes of sexually dimorphic structures
to either Female Pelvis or Male Pelvis (or Female Body or Male Body). From inputs such as
PP12(Uterus, Female Pelvis)
PP12(Prostate Gland, Male Pelvis)
PP1 assertions linking Pelvis (or Body) to the classes of sexually dimorphic structure can be generated via Table
PP1(Prostate Gland, Pelvis).
Thus, given this kind of mechanism for automatically generating PP1 and PP2 assertions from PP12 and Is_a as-
sertions and given also a rich enough classification system17, the curators of a biomedical ontology need only
manually input a portion of its PP12 and Is_a assertions to derive a full range of distinct PP1, PP2, and PP12 asser-
Of course, displaying at once all PP1, PP2, and PP12 assertions involving a given class is probably impractical. It
would also be redundant since for each assertion of the form PP12(A, B), we have also (via theorems (ClT26) –
(ClT27)) both PP1(A, B) and PP2(A, B). (For example, PP12(Urinary Bladder, Pelvis) entails the weaker asser-
tions: PP1(Urinary Bladder, Pelvis) and PP2(Urinary Bladder, Pelvis).) We suggest that PP1, PP2, and PP12 as-
sertions be displayed in separate (and clearly distinguished) modalities of the interface’s parthood graphs or ta-
bles of assertions. Examples illustrating some differences in the kind of information that would be embodied in
separate PP12, PP1, and PP12 graphs are given in Figure 6.
17 More precisely, the classification needs to satisfy the following two conditions: i) whenever PP1(A, B) holds but PP12(A, B) does NOT hold,
there is either some class C such that either C Is_a B and PP12(A, C) or A Is_a C and PP12(C, B); ii) whenever PP2(A, B) holds but PP12(A, B) does
NOT hold, there is some class D such that either B Is_a D and PP12(A, D) or D Is_a A and PP12(D, B).
Figure 6: Separate graphs for the PP12, PP1, and PP2 relations
The same kind of strategy can be used to input, derive, and display assertions involving R1, R2, and R12 class rela-
tions for other underlying relations R. For example, it can be used for assertions involving the broad BIT+Cl lo-
cation relations, Loc-In1, Loc-In2, and Loc-In12, or for assertions involving clearly distinguished versions of ei-
ther the FMA’s or GALEN’s containment relations. With a wide collection of class relations, more complex in-
ference rules can be implemented for automatically generating assertions. For example, Table 2 can be used to
generate further assertions from combinations of class parthood and class location assertions.
We have focused in this section on the advantages to be gained and the problems to be avoiding by including
clearly distinguished R1, R2, and R12 versions of parthood and location relation in a biomedical ontology. But we
have also seen other kinds of ambiguities in the FMA’s and GALEN’s uses of their parthood and containment
relations. For example, we saw that GALEN uses the same relation, isNonPartativelyContainedIn, both as a
class-level surround relation and as class-level location relation (in the sense of the Loc-Ini). We have also seen
that in some cases the FMA seems to use contained_in to mean: is-at-some-times-contained-in (as in Bolus of
Food contained_in Lumen of Esophagus). These kinds of ambiguities are undesirable and may, like the R1, R2,
R12 ambiguities, obstruct the development of robust reasoning mechanisms. A more complex extension of
BIT+Cl, which has a temporal component as well as a wider range of spatial relations, can be used to disambigu-
ate such diverse uses of the containment relations and also to analyze other types of relations (e.g. adjacency and
continuity) which are used in the FMA and GALEN.
Finally, we note that some of the BIT+Cl relations have no counterparts in the FMA or GALEN. We suggest
that biomedical ontologies consider expanding their collection of spatial inclusion relations so that they can in-
clude more anatomical information. For example, with PCoin12, the ontologies could assert:
PCoin12(Esophagus, Superior Mediastinal Space)
(every esophogus partially coincides with some superior mediastinal space and every superior mediastinal space
partially coincides with some esophagus)
PCoin12(Esophagus, Posterior Mediastinal Space)
(every esophogus partially coincides with some posterior mediastinal space and every posterior mediastinal space
partially coincides with some esophagus)
and so on. With O12, the ontologies could assert:
O12(Bony Pelvis, Vertebral Column)
(every bony pelvis overlaps some vertebral column and every vertebral column overlaps some bony pelvis)
Adding these particular relations to the ontologies would be especially advantageous since they have strong in-
ferential ties to the PP1, PP2, PP12, Loc-In1, Loc-In2, and Loc-In12 relations which correspond roughly to the
FMA’s and GALEN’s parthood and containment relations. For example, since the FMA has the assertions Sa-
crum part_of BonyPelvis and Sacrum part_of Vertebral Column and GALEN has analogous assertions stated
in terms of isDivisionOf18, O12(Bony Pelvis, Vertebral Column) could be inferred from information already in
the ontologies once the different versions of their parthood relations are clearly distinguished. Similarly,
PCoin12(Esophagus, Superior Mediastinal Space) can be derived in BIT+Cl from PP12(T4 Segment of Esopha-
gus, Esophagus) and Loc-In12(T4 Segment of Esophagus, Superior Mediastinal Space). In this way, given un-
18 Note that GALEN uses the term “SpinalColumn” instead of “VertebralColumn”.
ambiguous class parthood and containment assertions and strong automatic assertion generation capabilities,
many overlap and partial coincidence assertions could be generated without additional manual input.
6. Conclusions and Further Work
A central goal in artificial intelligence is to create ontologies which encode the general background knowledge
needed for organizing and using data in a specific domain such as medicine, biology, or geography. For these
domain ontologies to function as general references, they must be robust in the sense that they can be used in
different contexts by users with different kinds of expertise and different objectives. In particular, it should be
possible for users to integrate data organized in terms of a domain ontology with data organized according to a
different system. The domain ontologies should also be expandable – we should be able to add content or
stronger inference mechanisms without having to restructure the entire ontology.
To achieve these goals, it is crucial that the creators of an ontology organize the terms in their ontology in a
clear and systematic way and that all relational terms are linked to a formal theory which makes the logical prop-
erties of the relations explicit. Our investigation has shown that the spatial relational terms used to organize the
anatomical content of the FMA and GALEN are not clearly defined and that often the same relational term is
used for relations with significantly different logical properties. As a result, some assertions in these ontologies
are ambiguous and it is not obvious how to integrate anatomical information from the FMA with anatomical in-
formation in GALEN.19 We have also seen that the failure to distinguish different class-level relations obscures
the logical structure of the information embodied in these ontologies and limits possibilities for consistent
We have proposed Basic Inclusion Theory for Class (BIT+Cl) as a first-order logical theory in which different
class-level parthood and location relations can be clearly distinguished. The theory we develop here builds on
previous work [3, 5, 19]. We go beyond this earlier work in distinguishing an interconnected group of parthood
and location relations among individuals which are used to formally define corresponding class relations. We
have also investigated in much greater detail the logical properties of the relations introduced in our formal the-
ory and their correspondence with the relational terms of the FMA and GALEN.
Our approach can, in turn, be extended by strengthening the spatial component (BIT) of BIT+Cl. BIT can be
strengthened either through the addition of further restrictions on the relations already included in BIT (along the
lines suggested in Subsection 2.3 of this paper) or through the introduction of further relations. Further formal
relations are necessary for giving a full analysis of both the FMA’s and GALEN’s containment relations as well
as an analysis of other relations such as continuous_with or boundary_of. Another important area for further
research is the introduction of time dependent spatial relations (along the lines sketched in ) which can be
used in, e.g. developmental anatomy, to describe intermittent or evolving spatial relations between the instances
of two classes.
Finally, in order to link the assertions of canonical anatomy to either general descriptions of aberrant physical
structures or to descriptions of individual human beings’ particular body structures, some mechanism must be
introduced for handling abnormal anatomical structures. We have suggested (in Section 3.1) that this might be
done by distinguishing between normal and abnormal instances of a given class. Another promising approach is
to develop a version of BIT+Cl, not in standard first-order predicate logic, but in a non-monotonic logic .
Instead of forcing us to make a context-independent distinction between normal and abnormal instances of a
given class, a non-monotonic logic would allow us to over-ride certain general background assumptions (in par-
ticular, some of the assertions of canonical anatomy) when relevant information is provided about a specific indi-
vidual or group of individuals (e.g that a given individual has a lung tumor or has had a hysterectomy).
19 Note that obstacles to integration stemming from the use of unclear class-level relations are compounded by the mutually inconsistent classifica-
tion schemes adopted by these ontologies. For example, in the FMA Mediastinum is a subclass of Material Physical Anatomical Entity and in
GALEN Mediastinum is a subclass of Body Space.
We are very grateful for the helpful comments of Werner Cuesters, José Leonardo V. Mejino Jr., Fabian Neu-
haus, Jeremy Rogers, Barry Smith, and two anonymous reviewers. Thomas Bittner’s and Maureen Donnelly’s
work on this paper has been supported by the Wolfgang Paul Program of the Alexander von Humboldt Founda-
tion, the Network of Excellence in Semantic Interoperability and Data Mining in Biomedicine of the European
Union, and the project Forms of Life sponsored by the Volkswagen Foundation.
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