=Paper=
{{Paper
|id=Vol-222/paper-6
|storemode=property
|title=The Qualitative and Time-Dependent Character of Spatial Relations in Biomedical Ontologies
|pdfUrl=https://ceur-ws.org/Vol-222/krmed2006-p06.pdf
|volume=Vol-222
|dblpUrl=https://dblp.org/rec/conf/krmed/BittnerG06
}}
==The Qualitative and Time-Dependent Character of Spatial Relations in Biomedical Ontologies==
https://ceur-ws.org/Vol-222/krmed2006-p06.pdf
KR-MED 2006 "Biomedical Ontology in Action"
November 8, 2006, Baltimore, Maryland, USA
The qualitative and time-dependent character of spatial relations
in biomedical ontologies
Thomas Bittner1,3,4 and Louis J. Goldberg2,3
1
Departments of Philosophy and Department of Geography,
2
Departments of Oral Biology and Oral Diagnostic Sciences, School of Dental Medicine,
3
New York State Center of Excellence in Bioinformatics and Life Sciences
4
National Center of Geographic Information and Analysis (NCGIA)
State University of New York at Buffalo
{bittner3,goldberg}@buffalo.edu
Abstract in anatomical ontologies such as the FMA. It
The formal representation of mereological aspects is impossible to quantitatively describe aspects
of canonical anatomy (parthood relations) is rela- of shape and spatial arrangement of canonical
tively well understood. The formal representation anatomy. There is too much variation between
of other aspects of canonical anatomy like connect- the actual shapes and metric arrangements of par-
edness relations between anatomical parts, shape ticular structures among particular human beings.
and size of anatomical parts, the spatial arrange- Moreover it is the very nature of many anatomical
ment of anatomical parts within larger anatom- structures to change in shape and spatial arrange-
ical structures are, however, much less well un- ment over time: the heart beats, the jaw opens
derstood and only partial represented in compu- and closes, etc.
tational anatomical ontologies. In this paper we Qualitative representations of canonical anatomy
propose a methodology of how to incorporate this take advantage of the fact that despite the vari-
kind of information into anatomical ontologies by ations and changes in size, shape, distance, and
applying techniques of qualitative spatial represen- spatial arrangement, at the gross anatomical level,
tation and reasoning from Artificial Intelligence. all normal instances of the same biological species
As a running example we use the human temporo- are qualitative copies of each other. In all canoni-
mandibular joint (TMJ). cal anatomical structures certain parts need to be
present. These parts need to have certain qualita-
INTRODUCTION tive shape features (convex parts, concave parts,
other landmark features, etc.), their size must be
Anatomical ontologies are formal representations
within certain limits, and certain qualitative re-
of facts about the major parts of anatomical struc-
lations need to hold between those parts: some
tures, the qualitative shapes of those parts, and
parts are connected to others, some part are dis-
qualitative relations between them [19, 13, 30].
connected from others, some parts (like articular
The formal representation of mereological aspects
discs) need to be between other parts (like the
of canonical anatomy (parthood relations) is rela-
bones in synovial joints) etc.
tively well understood [16, 31, 13], and has been
implemented in computational medical ontologies In this paper we give an overview of the most im-
like the FMA [23], GALEN [22], and SNOMED portant of those relations. We also demonstrate
[32]. On the other hand, the formal representation how the changes in shape and arrangement can
of other aspects of canonical anatomy like connect- be specified using qualitative spatial relations. In
edness relations between anatomical parts, shape addition, we claim that most pathological cases
and size of anatomical parts, the spatial arrange- can also be characterized and distinguished from
ment of anatomical parts within larger anatomi- non-pathological cases in terms of qualitative re-
cal structures are less well understood and only lations: there may be too many or too few parts,
partially represented in computational anatomical parts that are supposed to be connected are dis-
ontologies. In this paper we propose a methodol- connected, parts that are supposed to be between
ogy of how to incorporate this kind of information other parts fail to be so, etc.
into anatomical ontologies. Qualitative representation of, and reasoning about
We stress here the importance of recognizing complex systems has a long tradition in Artificial
the qualitative nature of all facts represented Intelligence [34, 5, 10]. Cohn and Hazarika [8]
47
stress that the essence of qualitative representa- characterize the spatial arrangement of the
tions is to find ways to represent continuous prop- parts within the structures;
erties of the world by discrete systems of symbols.
As Forbus [14] points out, one can always quantize 3. Refine ordering relations between parts by iden-
something continuous, but not all quantizations tifying anatomical landmarks and by using land-
are equally useful because the distinctions made marks as a frame of reference;
by a quantization must be relevant for the kind of
reasoning performed. This is where formal ontol- 4. Specify qualitative distance relations between
ogy comes into play [29]. It will be an important landmarks to qualitatively characterize shape
aspect of this paper to show how to discretize con- and arrangement of the parts.
tinuous domains in such a way that ontologically
significant properties are preserved. We will discuss each step below in sequence and
For example, to qualitatively model the behav- use the human temporomandibular joint (TMJ)
ior of water at different temperatures the continu- as a running example. We go into a detailed dis-
ous domain of temperature is discretized by intro- cussion of how existing techniques of qualitative
ducing landmark values: temperature landmark 1 spatial representation and reasoning from Artifi-
(TLM1) the temperature at which water changes cial intelligence can be used and extended to for-
from its solid state to its liquid state and (TLM2) mally and qualitatively represent the mereotopol-
the temperature where water changes from its liq- ogy of anatomical structures, the shape and size of
uid state to being a gas. These landmark values anatomical parts, and the spatial arrangement of
bound intervals: for example, (TI1) the interval anatomical parts within larger anatomical struc-
of temperatures at which water is solid, (TI2) the tures. The methods we present here we believe
interval of temperatures at which water is liquid, will provide the foundations for the next genera-
and (TI3) the (half open) interval at which water tion of anatomical ontologies.
is a gas. In a qualitative model the behavior of wa-
ter at different temperatures is described only by ANATOMICAL PARTS AND
referring to the landmark values and the intervals MEREOTOPOLOGICAL
bounded by those values. RELATIONS
An important point is that the landmarks are not
chosen arbitrarily. The landmarks represent sig- Parthood relations
nificant changes in the domain at hand, while At the most basic level of the study of the canon-
within the intervals between landmarks no signif- ical structure of the TMJ we consider its anatom-
icant changes occur. Thus qualitative representa- ical parts. Anatomical parts here means, maxi-
tions focus on ontologically salient features. For mally connected parts of non-negligible size (thus
many purposes this qualitative representation of cells and molecules are parts of anatomical struc-
water at different temperatures will be sufficient. tures but not anatomical parts). At this gross
For example, in order to transport bottled water anatomical granularity we will distinguish two
from one place to another the exact temperature of kinds of anatomical parts: material parts and cav-
the water is irrelevant as long as it does not freeze ities. The material anatomical parts of the TMJ
or change to its gas state since in both cases the at the gross anatomical level of granularity accord-
bottled water will destroy their containers. ing to [18] are depicted in Figure 1, which shows,
in a sagittal section through the middle of the
We propose the following methodology for condyle, a TMJ in closed (a) and open (b) jaw
building qualitative representations of canonical position: temporal bone (1), head of condyle (2),
anatomical structures that preserve ontologically articular disc (3), posterior attachment (4), lat-
significant distinctions: eral pterygoid muscle (5). Immaterial anatomical
parts (cavities) are the superior and inferior syn-
1. Specify and classify the major canonical parts ovial cavities, which are depicted as white spaces
of the structure at hand and establish canonical above and below the articular disc and the poste-
mereotopological (parthood and connectedness) rior attachment. Here we will focus on material
relations between them; parts. For a discussion of immaterial anatomical
parts see [12, 26, 19].
2. Identify ordering relations between the ma- A clear understanding of the number and kinds
jor parts anatomical structures to qualitatively of canonical parts of an anatomical structure is
48
at t but x and y do not overlap at t. Externally
(1) connected regions share boundary points but no
(3) interior points. Objects x and y are disconnected
(2) (5) at time t if and only if x and y are not connected
(4) at t.
We introduce connectedness as a time-dependent
(b)
relation since anatomic structures can be con-
nected to different (parts of) structures at differ-
Figure 1: Drawings of (a) the major parts of a ent times. As depicted in Figure 1(a), at time t1
TMJ in the jaw closed position and (b) the major the articular disc is (externally) connected to the
parts of the same TMJ in the jaw open position. fossa (a fiat part1 of the temporal bone). At time
t2 , as depicted in Figure 1(b) the articular disc is
connected to the articular eminence (another fiat
critical for identifying non-canonical (and poten- part of the temporal bone).
tially pathological) parts such as tumors. More- The following topological relations hold between
over, without a clear understanding of the number the five major parts of the TMJ depicted in Fig-
of canonical parts it is not possible to recognize the ures 1(a) and (b): the temporal bone (1) is ex-
absence of certain parts. In the remainder of this ternally connected to the posterior attachment
paper we refer to individual anatomical structures (4) and to the lateral pterygoid muscle (5). The
and their material anatomical parts as objects. condyle (2) is externally connected to the poste-
Parthood is a ternary relation (a relation with rior attachment (4) and to the lateral pterygoid
three arguments) that holds between two objects muscle (5). The articular disc (3) is externally
x and y and a time instant t. Parthood is a time- connected to the posterior attachment (4) and the
dependent relation since anatomic structures can lateral pterygoid muscle (5).
have different parts at different times. For exam-
ple, in the course of their transition from children Permanent parthood and
to adults, it is normal for people to have differ- connectedness
ent teeth at different times. See, for example, [27] Consider the relation of external connectedness
for axiomatic formalizations time-dependent part- between the articular disc and the temporal bone.
hood. Clearly, at every time t the articular disc is exter-
In terms of parthood we define the relations of nally connected (in external contact) to some part
proper parthood and overlap. Object x is a proper of the temporal bone. However at different times
part of object y at t if and only if x is a part of y the articular disc is externally connected (in ex-
at t and y is not part of x at t. For example, at ternal contact) to different parts of the temporal
time t the head of Joe’s condyle is a proper part bone. In Figure 1 (a) the articular disc is exter-
of his condyle. Object x overlaps object y at time nally connected (in external contact) to the fossa,
t if and only if there is an object z such that z is while in Figure 1 (b) the articular disc is exter-
part of x at t and z is part of y at t. If x is a nallhy connected (in external contact) to the ar-
(proper) part of y at t then x and y overlap at t. ticular eminence (another fiat part of the temporal
Thus, at time t Joe’s condyle and the head of his bone).
condyle overlap. It is important to make explicit that the connect-
edness relation between the articular disc and the
Connectedness relations temporal bone is different from the connectedness
The ternary relation of connectedness holds be- relation between the articular disc the posterior
tween two objects x and y at a time instant t. attachment and the lateral pterygoid muscle: at
Intuitively, x is connected to y at t if and only all times at which the articular disc is connected
if x and y overlap at t or x and y are in direct to the posterior attachment it is connected to the
external contact at t. Two regions are connected same part of the posterior attachment and simi-
at t if and only if they share at least a bound- larly for the lateral pterygoid muscle. The rela-
ary point at t (they may share interior points at tion between articular disc and posterior attach-
t). For a discussion of the wide range of possible ment is a relation of constant or permanent con-
formalizations see [33]. 1
A fiat part is a part which boundaries are (partly)
Objects x and y are externally connected at time t the result of human demarcation and do not corre-
if and only if x and y are in direct external contact spond to discontinuities in reality [28].
49
nection (articular disc and posterior attachment bone (1), the articular disc (3), the posterior at-
are ‘glued’ together by direct connective tissue at- tachment (4), and the lateral pterygoid muscle (5)
tachments). On the other hand the relationship are constant proper parts of the TMJ.
between articular disc and temporal bone is such The solid edges in the graph in Figure 2(a) rep-
that both are externally connected (in external resent constant connectedness relations between
contact) but the articular disc has the freedom to parts of the TMJs depicted in Figure 1 (a) and (b):
slide along the surface of the bone.2 at all times at which the TMJ as a whole exists the
condyle (2) is (externally) connected to the pos-
terior attachment (4) and to the lateral pterygoid
(1)
(1) muscle (5). By contrast, a (with respect to time)
different connectedness relation bolds between ar-
(3)
(4)
(3) (5)
(2)
ticular disc (3) and the temporal bone (1) and the
(4) articular disc and the head of the condyle (2): the
(5)
disc is externally connected to different parts of
(2)
(a) (b) the temporal bone and the head of the condyle at
different times. In the graph in Figure 2(a) this is
Figure 2: (a) Graph structure which represents represented by dotted edges between the respec-
the relations of external connectedness between tive nodes.
the major parts of the TMJ, (b) TMJ with ar-
ORDERING RELATIONS
ticular disc not positioned between condyle and
temporal bone. BETWEEN EXTENDED
OBJECTS
We define the following constant mereotopological Mereotopology alone is not powerful enough to
relations: Object x is a constant part of object y sufficiently characterize the important properties
if and only if whenever y exists, x is a part of y. of TMJs. Consider the graph in Figure 2(a),
Object x is a constant proper part of object y if which is a graph-theoretical representation of the
and only if whenever y exists, x is a proper part of mereotopological properties of the TMJs depicted
y. Object x is a constantly connected to object y in Figures 1(a), 1(b), and 2(b). The fact that the
if and only if whenever y exists, x is connected to TMJs depicted in the three figures have the same
y. Object x is a constantly externally connected graph-theoretic representation shows that in terms
to object y if and only if whenever y exists, x is of mereotopological properties we cannot distin-
externally connected to y. Object x is a constantly guish the TMJs in Figures 1(a), 1(b), and 2(b).
disconnected from object y if and only if whenever Obviously it is critical to distinguish the TMJ in
y exists, x is disconnected to y. Figure 2(b) from the TMJs in Figures 1(a) and
Consider Figure 2 (a). Every part of the TMJs 1(b). It is the purpose of the articular disc in
in Figure 1 (a) and (b) is topologically equivalent a TMJ to be between the condyle and temporal
to a filled circle which is indicated by the corre- bone at all times. If we take the ordering relation
sponding labels of the dots in Figure 2. Moreover, of betweenness into account then the TMJs in Fig-
the nodes (the labeled circles) in the graph repre- ures 1(a) and 1(b) can be distinguished from the
sent constant proper parts of the TMJ: at all times clearly pathological TMJ in Figure 2(b) where the
at which the TMJ as a whole exists, the condyle posterior attachment is between the condyle and
(2) is a proper part of it. Similarly the temporal the temporal bone and not the articular disc.
Ordering relations like betweenness describe the
2
Strictly speaking, this ability to slide is due to location of disjoint objects relatively to one other.
the fact that the articular disc is separated from the Besides betweenness, ordering relations include:
temporal bone by a film of fluid which fills the su- left-of, right-of, in-front-of, above, below, behind,
perior synovial cavity. As stated previously, for the
purpose of this paper we will not consider cavities or etc. The science of anatomy has developed a whole
holes, and so will consider that the articular disc is set of ordering relation terms to describe the ar-
effectively free to slide to various positions along the rangement of anatomical parts in the human body:
surface of the temporal bone. Notice, however, that superior, inferior, anterior, posterior, lateral, me-
we could introduce a relation of adjacency. We would dial, dorsal, ventral, rostral, proximal, distal, etc.
then have to distinguish between constant adjacency
and temporary adjacency in the same way we distin- The FMA, for example, has an ‘orientation net-
guish constant external connectedness and temporary work’ in which these kinds of relations are repre-
external connectedness. sented [23].
50
Unfortunately, ordering relations between spa- points are landmarks of anatomical structures of
tially extended objects are difficult to formalize. a given kind if and only if:
As [11] points out in her treatment of relation of
1. They exist as parts of every anatomical struc-
betweenness: ‘The problem with trying to char-
ture of that kind;
acterize the betweeness relation on extended ob-
jects is that we typically use the betweeness re- 2. They are critical for the normal function of all
lation only on objects that have fairly uniform anatomical structures of that kind.
shapes and are nearly the same size. It is unclear
Thus the salient points LM1-LM6 in Figure 3 are
whether or not the betweeness relation should hold
anatomical landmarks of temporal bones of nor-
in certain cases involving irregularly shaped ob-
mal human TMJs, since (a) they exist as parts of
jects and differently sized objects.’ Similar prob-
every temporal bones of a normal human TMJ and
lems face attempts to formalize qualitative direc-
(b) they are important for the function of a human
tion relations between spatially extended objects,
TMJ as a whole. Consequently, independently of
e.g., [20]. Similarly it is very difficult to qualita-
the normal variations between the actual shape
tively describe distances between extended objects
of temporal bones in different human beings, all
particularly if they are of different size and shape,
normal temporal bones will have the landmarks
e.g., [36, 35].
LM1-LM7 as depicted in Figure 3.
LANDMARKS Qualitative distances between
To avoid problems that occur when describing or- landmarks
dering relations between extended objects we will Although normal temporal bones in human TMJs
choose a different approach: we will characterize will have the landmarks LM1-LM7, the particular
shape, extent, and spatial arrangement of anatom- metric properties like the actual height of the max-
ical structures and their anatomical parts using imum, the actual depth of the minimum, as well
(point-like) anatomical landmarks [6] and qualita- as their actual distance, will vary from individual
tive ordering relations between the landmarks. to individual.
Consider the landmarks of the temporal bone de-
Landmarks of anatomical structures picted in Figure 3. Rather than quantitatively
Intuitively, anatomical landmarks are special characterizing shape differences in terms of coor-
salient points on the surface of anatomical struc- dinate differences among the landmarks, we can
tures or their anatomical parts [6]. Consider the characterize the shape differences qualitatively by
temporal bone in Figure 3. Salient points on the specifying qualitative distance relations between
inferior surface of the temporal bone are local min- those landmarks. Consider, for example, the
ima (LM3, LM7), local maxima (LM1, LM5) as anatomical landmarks LM1 and LM3. In Figure 3
well as points at which changes from convexity to the coordinate difference along the anterior (hori-
concavity occur (LM2, LM4, LM6). zontal) axis is smaller than the coordinate differ-
ence along the rostral (vertical) axis. Similarly
the coordinate difference between LM3 and LM5
along the anterior axis is roughly twice as large as
the coordinate difference along the rostral axis.
Since all TMJs will have the same landmarks on
LM3 their temporal bones (assuming a certain degree
LM7
LM4 LM6 of anatomical normality), we can classify TMJs
LM5 according to qualitative coordinate differences be-
LM2 tween their landmarks. There are many ways of
R doing this. Here we only discuss some examples to
demonstrate the power of the qualitative method-
A
LM1
ology. In particular we focus on the landmarks
LM1, LM3, and LM5.
Given a coordinate system3 existing coordinate
Figure 3: Landmarks on Joe’s temporal bone.
3
We do not need the coordinate system for mea-
surement. We only use it to distinguish coordinate dif-
However not all salient points on the surface of a ferences in anterior (horizontal) direction (δh) from co-
given anatomical structure are landmarks. Salient ordinate differences in rostral (vertical) direction (δv).
51
differences between LM1 and LM3 along the an- Qualitative directions and orientation
terior axis (δa13 ) and along the rostral axis (δr31 ) relations between landmarks
can be used to distinguish the following cases: There exist a variety of approaches to qualitatively
δa13 = δr31 , δa13 < δr31 , and δa13 > δr31 . Here represent angles between landmarks and to use
δa13 = δr31 means that δa13 is as large as δr31 , landmarks as origins for qualitative frames of ref-
δa13 < δr31 means that δa13 is smaller than δr31 , and erences. For example, the landmark ‘LM’ in Fig-
δa13 > δr31 means that δa13 is larger than δr31 . No- ure 4(a) could serve as the origin of the qualitative
tice that this classification is jointly exhaustive and frame of reference in Figure 4(b). We then could
pairwise disjoint. That is, for any possible constel- specify the location of anatomical landmarks of
lation of the anatomical landmarks LM1 and LM3 the heart within this frame of reference.
exactly one of those relations holds. In Figure 3 Most of the approaches to qualitative orientation
the rostral coordinate difference between LM1 and and directions also incorporate qualitative dis-
LM3 is larger than the anterior coordinate differ- tance relations like close, near, far, etc. (where
ence between LM1 and LM3, i.e., δa13 < δr31 . close, near, and far roughly correspond to the re-
Of course we can in addition classify the ante- lations ∼, <, and – see for example, [7, 4]
rior and rostral coordinate differences between the for details). In Figure 4 we then could say that
landmarks LM3 and LM5 in the same way. If we all anatomical landmarks of the heart are near
take both classifications together then the follow- and in front with respect to the frame of refer-
ing nine combinations are combinatorially possi- ence which is centered at the landmark LM. More
ble: sophisticated ways of representing qualitative or-
R∈ der relations between landmarks were proposed in
{=, <, >} 1 2 3 4 5 6 7 8 9 [15, 24, 25].
δa13 R δr31 = = = < < < > > >
δa35 R δr53 < > = < > = < > = front, far
Any possible constellation of LM1, LM2, and LM3 front near
is characterized by exactly one column in this ta- left
near
left
close
right right
near
ble. In Figure 3 we have δa13 < δr31 and δa35 > δr53 . near far
which corresponds to column 5 in the above table. back near
Since this classification is exhaustive we now can back far
analyze which of the nine possibilities are nor- (a)
LM
(b)
mal and which are pathological or which correlate
with certain clinical symptoms. This analysis may Figure 4: (a) a radiographic section taken through
show that distinguishing nine cases is insufficient a human thorax. Arrows point to the heart. LM,
to make the necessary distinction to distinguish Is a point in the center of the spinal cord. (b) qual-
normal anatomy form various kinds of pathologies. itative ordering and qualitative distance relations
In this case we have three options: (a) take more according to Hernandez [17].
landmarks into account; (b) distinguish more re-
lations; (c) do both (a) and (b).
Consider option (b) instead of distinguishing three
APPROXIMATE LOCATION IN
relations =, <, and > we could add two more re-
lations: and interpreted as much smaller FRAMES OF REFERENCE
and much bigger respectively. Another way of dis- There are many ways to represent approximate lo-
tinguishing more relations would be to refine > by cation in qualitative frames of references. (See, for
distinguishing twice as big, three times as big, etc. example [3].) Here we discuss a specific technique
There are no limits to this method provided the which is useful in the context of our TMJ example.
resulting set of relations is jointly exhaustive and Consider the boundary of Joe’s temporal bone as
pairwise disjoint. depicted in Figure 3. Topologically, the boundary
Notice that it might be more realistic to replace is a one-dimensional curve. Since the landmarks
the identity relation = by the relation ∼, were LM1-LM7 are points on this curve, each landmark
δa ∼ δr means that δa is roughly as large as δr. is a boundary of at least one interval (a one-piece
The exact definitions of the relations ∼, , and part of the underlying curve). For example, in
are not trivial and their formalization is beyond Figure 3 the landmarks LM2 and LM3 bound the
the scope of this paper. For discussions of existing interval which is formed by the part of the curve
approaches see [21, 9, 7, 4]. between them. We use the landmarks that bound
52
a given interval to refer to this interval. For exam- tions which hold at time t between the projection
ple, we write L2L3 to refer to the interval bounded of the articular disc at t and the intervals bounded
by LM2 and LM3 in Figure 3. by the landmarks. These mereotopological rela-
In our mereotopological framework we can repre- tions at time t1 and t2 can be summarized as:
sent the topological relations between the intervals
Joe’s
formed by the anatomical landmarks of Joe’s tem- disc L1L2 L2L3 L3L4 L4L5 L5L6 L6L7
poral bone as: Interval L1L2 is constantly exter- t1 DC EC COV PO DC DC
nally connected to interval L2L3, interval L2L3 is t2 DC DC DC PO PO DC
constantly externally connected to interval L3L4,
The first row reads as DC(Prj(D, t1 ), L1L2, t1 ),
and so on.
EC(Prj(D, t1 ), L2L3, t1 ), . . . and similarly for the
LM3 LM6
LM7
LM3 LM6
LM7 second row.
LM4 LM5 LM4 LM5
Consider the images shown in Figures 6(a) and (b)
LM2 LM2 which depict the relative location of Joe’s condyle
with respect to his temporal bone at times t1 and
R R t2 respectively. Figure 6(a) corresponds to Figure
A A
LM1 LM2 LM3 LM4 LM5 LM1 LM2 LM3 LM4 LM5 1(a) and Figure 6(b) corresponds to Figure 1(b).
(a) (b) In the same way we projected Joe’s disc onto the
boundary of his temporal bone to identify an inter-
Figure 5: Relations between articular disc and val that can be related to the intervals bounded by
landmark intervals of the temporal bone at times the landmarks LM1-LM7, we can project the head
t1 (a) and t2 (b). of his condyle onto the boundary of his temporal
bone as indicated by the dotted lines in Figures 6
(a) and (b).
Consider Figures 5(a) and (b) which depict the rel-
ative location of Joe’s articular disc with respect LM7 LM7
LM3 LM6
to his temporal bone at times t1 and t2 respec- LM4 LM5 LM3
LM4 LM5
LM6
tively. Figure 5(a) corresponds to Figure 1(a) and LM2 LM2
both show Joe’s TMJ in the jaw closed position.
Similarly, Figure 5(b) corresponds to Figure 1(b) R R
and both show Joe’s TMJ in the jaw open posi- LM1
A
LM1
A
tion. On the bottom of both images in Figure 5 the (a) (b)
projection of Joe’s articular disc onto the bound-
ary of his temporal bone is depicted. From this Figure 6: Mereotopological relations between the
point on, we will write Prj(D, t) to refer the in- head of the condyle and landmark intervals of the
terval that is the projection of Joe’s articular disc temporal bone at times t1 (a) and t2 (b).
on the boundary of his temporal bone in a sagittal
section through the middle of his condyle at time As in the case of Joe’s disc, at every time t we can
t. specify the location of the head of Joe’s condyle
The interval Prj(D, t) stands in mereotopological with respect to the landmarks of his temporal bone
relationships to the intervals bounded by the land- it terms of the relations which hold at time t be-
marks LM1-LM7. For example, at time t1 the tween the projection the head of the condyle at t
projection of Joe’s articular disc completely covers and the intervals bounded by the landmarks. The
the interval L3L4, i.e., COV(P rj(D, t1 ), L3L4, t1 ). spatial relations at time t1 and t2 can be summa-
In other words the interval L3L4 is a part rized as:
of the projection of Joe’s articular disc, i.e.,
Joe’s
PartOf(L3L4, Prj(D, t1 ), t1 ). Notice that at condyle L1L2 L2L3 L3L4 L4L5 L5L6 L6L7
time t2 the projection of Joe’s articular disc t1 DC EC PO DC DC DC
and the interval L3L4 are disconnected, i.e., t2 DC DC DC PO PO DC
DC(L3L4, Prj(D, t2 ), t2 ).4 If we use C to denote the head of Joe’s condyle
Thus at every time t we can specify the location then the first row reads as DC(Prj(C, t1 ), L1L2, t1 ),
of Joe’s articular disc with respect to the land- EC(Prj(C, t1 ), L2L3, t1 ), . . . , and similarly for the
marks of his temporal bone in terms of the rela- second row. Notice that the table with the rela-
4
For details of the exact definitions of the relations tions of Joe’s articular disc corresponds nicely to
between the intervals see [1, 2]. the table with the relations of the head of Joe’s
53
condyle, i.e., the articular disc is at both times be- 2. The strict distinction of time-dependent and
tween the head of the condyle and the temporal time-independent relations;
bone.
3. The identification of anatomical landmarks for
Clearly, for every possible location of an articular the representation of the shape of anatomical
disc in a TMJ with respect to the temporal bone parts and the spatial arrangement of anatomical
of this TMJ there is a unique sequence of rela- structures;
tions similar to those in the table of Joe’s disc.
Similarly, for every possible location of the head 4. The identification of sets of jointly exhaustive
of a condyle in a TMJ with respect to the tempo- and pairwise disjoint relations to describe rela-
ral bone of this TMJ there is a unique sequence tions between anatomical parts and anatomical
of relations similar to those in the table of Joe’s landmarks;
condyle. Moreover, since we have, (i) the same
5. The establishment of landmarks and qualita-
anatomical landmarks on the temporal bones of
tive distinctions that reflect the ontologically
every normal TMJ and, (ii) there are only a fi-
significant aspects of the canonical anatomy of
nite number of mereotopological relations that can
biomedical structures as well as relevant patho-
hold between two intervals, we can therefore, com-
logical cases.
pose two finite tables: one table in which each row
corresponds to one anatomically possible location This methodology permits, in principle, the
of some articular disc with respect to the corre- exhaustive qualitative characterization of all
sponding temporal bone; a second table in which anatomically possible instantiations of anatomical
each row corresponds to one anatomically possible structures. These then can be classified as normal
location of the head of some condyle with respect or pathological and correlated with other clinical
to the corresponding temporal bone.5 Both tables findings.
together contain all possible combinations of lo- The discussion in this paper exclusively focused
cations of the head of a condyle and an articular on relations between particulars (Joe Doe’s TMJ).
disc with respect to the landmarks of a temporal It is well known that anatomical ontologies are
bone in any possible TMJ. Some of these combina- mostly about relations between universals or
tions we can classify as normal (among these are classes [31, 30]. However it is also well known that
the two tables above) others are pathological and relations between universals or classes are defined
again others will be anatomically impossible and in terms of relations between particulars [13].
thus can be ruled out.
Address for Correspondence
CONCLUSIONS Thomas Bittner, State University of New York, De-
The purpose of this paper is to show that there partment of Philosophy, 135 Park Hall, Buffalo (NY),
can be obtained, by following the methodology we 14260, USA
have presented here, a series of well understood
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