The formal representation of mereological aspects of canonical anatomy (parthood relations) is relatively well understood. The formal representation of other aspects of canonical anatomy like connectedness relations between anatomical parts, shape and size of anatomical parts, the spatial arrangement of anatomical parts within larger anatomical structures are, however, much less well understood and only partial represented in computational anatomical ontologies. In this paper we propose a methodology of how to incorporate this kind of information into anatomical ontologies by applying techniques of qualitative spatial representation and reasoning from Artificial Intelligence. As a running example we use the human temporomandibular joint (TMJ).
Anatomical ontologies are formal representations
of facts about the major parts of anatomical
structures, the qualitative shapes of those parts, and
qualitative relations between them [
We stress here the importance of recognizing the qualitative nature of all facts represented in anatomical ontologies such as the FMA. It is impossible to quantitatively describe aspects of shape and spatial arrangement of canonical anatomy. There is too much variation between the actual shapes and metric arrangements of particular structures among particular human beings. Moreover it is the very nature of many anatomical structures to change in shape and spatial arrangement over time: the heart beats, the jaw opens and closes, etc.
Qualitative representations of canonical anatomy take advantage of the fact that despite the variations and changes in size, shape, distance, and spatial arrangement, at the gross anatomical level, all normal instances of the same biological species are qualitative copies of each other. In all canonical anatomical structures certain parts need to be present. These parts need to have certain qualitative shape features (convex parts, concave parts, other landmark features, etc.), their size must be within certain limits, and certain qualitative relations need to hold between those parts: some parts are connected to others, some part are disconnected from others, some parts (like articular discs) need to be between other parts (like the bones in synovial joints) etc.
In this paper we give an overview of the most important of those relations. We also demonstrate how the changes in shape and arrangement can be specified using qualitative spatial relations. In addition, we claim that most pathological cases can also be characterized and distinguished from non-pathological cases in terms of qualitative relations: there may be too many or too few parts, parts that are supposed to be connected are disconnected, parts that are supposed to be between other parts fail to be so, etc.
Qualitative representation of, and reasoning about
complex systems has a long tradition in Artificial
Intelligence [
For example, to qualitatively model the behavior of water at different temperatures the continuous domain of temperature is discretized by introducing landmark values: temperature landmark 1 (TLM1) the temperature at which water changes from its solid state to its liquid state and (TLM2) the temperature where water changes from its liquid state to being a gas. These landmark values bound intervals: for example, (TI1) the interval of temperatures at which water is solid, (TI2) the interval of temperatures at which water is liquid, and (TI3) the (half open) interval at which water is a gas. In a qualitative model the behavior of water at different temperatures is described only by referring to the landmark values and the intervals bounded by those values.
An important point is that the landmarks are not chosen arbitrarily. The landmarks represent significant changes in the domain at hand, while within the intervals between landmarks no significant changes occur. Thus qualitative representations focus on ontologically salient features. For many purposes this qualitative representation of water at different temperatures will be sufficient. For example, in order to transport bottled water from one place to another the exact temperature of the water is irrelevant as long as it does not freeze or change to its gas state since in both cases the bottled water will destroy their containers. We propose the following methodology for building qualitative representations of canonical anatomical structures that preserve ontologically significant distinctions: 1. Specify and classify the major canonical parts of the structure at hand and establish canonical mereotopological (parthood and connectedness) relations between them; 2. Identify ordering relations between the major parts anatomical structures to qualitatively characterize the spatial arrangement of the parts within the structures; 3. Refine ordering relations between parts by identifying anatomical landmarks and by using landmarks as a frame of reference; 4. Specify qualitative distance relations between landmarks to qualitatively characterize shape and arrangement of the parts.
We will discuss each step below in sequence and use the human temporomandibular joint (TMJ) as a running example. We go into a detailed discussion of how existing techniques of qualitative spatial representation and reasoning from Artificial intelligence can be used and extended to formally and qualitatively represent the mereotopology of anatomical structures, the shape and size of anatomical parts, and the spatial arrangement of anatomical parts within larger anatomical structures. The methods we present here we believe will provide the foundations for the next generation of anatomical ontologies.
At the most basic level of the study of the
canonical structure of the TMJ we consider its
anatomical parts. Anatomical parts here means,
maximally connected parts of non-negligible size (thus
cells and molecules are parts of anatomical
structures but not anatomical parts). At this gross
anatomical granularity we will distinguish two
kinds of anatomical parts: material parts and
cavities. The material anatomical parts of the TMJ
at the gross anatomical level of granularity
according to [
A clear understanding of the number and kinds
of canonical parts of an anatomical structure is
(1)
(4)
critical for identifying non-canonical (and
potentially pathological) parts such as tumors.
Moreover, without a clear understanding of the number
of canonical parts it is not possible to recognize the
absence of certain parts. In the remainder of this
paper we refer to individual anatomical structures
and their material anatomical parts as objects.
Parthood is a ternary relation (a relation with
three arguments) that holds between two objects
x and y and a time instant t. Parthood is a
timedependent relation since anatomic structures can
have different parts at different times. For
example, in the course of their transition from children
to adults, it is normal for people to have
different teeth at different times. See, for example, [
In terms of parthood we define the relations of proper parthood and overlap. Object x is a proper part of object y at t if and only if x is a part of y at t and y is not part of x at t. For example, at time t the head of Joe’s condyle is a proper part of his condyle. Object x overlaps object y at time t if and only if there is an object z such that z is part of x at t and z is part of y at t. If x is a (proper) part of y at t then x and y overlap at t. Thus, at time t Joe’s condyle and the head of his condyle overlap.
The ternary relation of connectedness holds
between two objects x and y at a time instant t.
Intuitively, x is connected to y at t if and only
if x and y overlap at t or x and y are in direct
external contact at t. Two regions are connected
at t if and only if they share at least a
boundary point at t (they may share interior points at
t). For a discussion of the wide range of possible
formalizations see [
Objects x and y are externally connected at time t if and only if x and y are in direct external contact at t but x and y do not overlap at t. Externally connected regions share boundary points but no interior points. Objects x and y are disconnected at time t if and only if x and y are not connected at t.
We introduce connectedness as a time-dependent relation since anatomic structures can be connected to different (parts of) structures at different times. As depicted in Figure 1(a), at time t1 the articular disc is (externally) connected to the 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 part of the temporal bone).
The following topological relations hold between the five major parts of the TMJ depicted in Figures 1(a) and (b): the temporal bone (1) is externally connected to the posterior attachment (4) and to the lateral pterygoid muscle (5). The condyle (2) is externally connected to the posterior attachment (4) and to the lateral pterygoid muscle (5). The articular disc (3) is externally connected to the posterior attachment (4) and the lateral pterygoid muscle (5).
Consider the relation of external connectedness between the articular disc and the temporal bone. Clearly, at every time t the articular disc is externally connected (in external contact) to some part of the temporal bone. However at different times the articular disc is externally connected (in external contact) to different parts of the temporal bone. In Figure 1 (a) the articular disc is externally connected (in external contact) to the fossa, while in Figure 1 (b) the articular disc is externallhy connected (in external contact) to the articular eminence (another fiat part of the temporal bone).
It is important to make explicit that the
connectedness relation between the articular disc and the
temporal bone is different from the connectedness
relation between the articular disc the posterior
attachment and the lateral pterygoid muscle: at
all times at which the articular disc is connected
to the posterior attachment it is connected to the
same part of the posterior attachment and
similarly for the lateral pterygoid muscle. The
relation between articular disc and posterior
attachment is a relation of constant or permanent
con1A fiat part is a part which boundaries are (partly)
the result of human demarcation and do not
correspond to discontinuities in reality [
(5)
We define the following constant mereotopological relations: Object x is a constant part of object y if and only if whenever y exists, x is a part of y. Object x is a constant proper part of object y if and only if whenever y exists, x is a proper part of y. Object x is a constantly connected to object y if and only if whenever y exists, x is connected to y. Object x is a constantly externally connected to object y if and only if whenever y exists, x is externally connected to y. Object x is a constantly disconnected from object y if and only if whenever y exists, x is disconnected to y.
Consider Figure 2 (a). Every part of the TMJs in Figure 1 (a) and (b) is topologically equivalent to a filled circle which is indicated by the corresponding labels of the dots in Figure 2. Moreover, the nodes (the labeled circles) in the graph represent constant proper parts of the TMJ: at all times at which the TMJ as a whole exists, the condyle (2) is a proper part of it. Similarly the temporal 2Strictly speaking, this ability to slide is due to the fact that the articular disc is separated from the temporal bone by a film of fluid which fills the superior synovial cavity. As stated previously, for the purpose of this paper we will not consider cavities or holes, and so will consider that the articular disc is effectively free to slide to various positions along the surface of the temporal bone. Notice, however, that we could introduce a relation of adjacency. We would then have to distinguish between constant adjacency and temporary adjacency in the same way we distinguish constant external connectedness and temporary external connectedness. bone (1), the articular disc (3), the posterior attachment (4), and the lateral pterygoid muscle (5) are constant proper parts of the TMJ.
The solid edges in the graph in Figure 2(a) represent constant connectedness relations between parts of the TMJs depicted in Figure 1 (a) and (b): at all times at which the TMJ as a whole exists the condyle (2) is (externally) connected to the posterior attachment (4) and to the lateral pterygoid muscle (5). By contrast, a (with respect to time) different connectedness relation bolds between articular disc (3) and the temporal bone (1) and the articular disc and the head of the condyle (2): the disc is externally connected to different parts of the temporal bone and the head of the condyle at different times. In the graph in Figure 2(a) this is represented by dotted edges between the respective nodes.
Mereotopology alone is not powerful enough to
sufficiently characterize the important properties
of TMJs. Consider the graph in Figure 2(a),
which is a graph-theoretical representation of the
mereotopological properties of the TMJs depicted
in Figures 1(a), 1(b), and 2(b). The fact that the
TMJs depicted in the three figures have the same
graph-theoretic representation shows that in terms
of mereotopological properties we cannot
distinguish the TMJs in Figures 1(a), 1(b), and 2(b).
Obviously it is critical to distinguish the TMJ in
Figure 2(b) from the TMJs in Figures 1(a) and
1(b). It is the purpose of the articular disc in
a TMJ to be between the condyle and temporal
bone at all times. If we take the ordering relation
of betweenness into account then the TMJs in
Figures 1(a) and 1(b) can be distinguished from the
clearly pathological TMJ in Figure 2(b) where the
posterior attachment is between the condyle and
the temporal bone and not the articular disc.
Ordering relations like betweenness describe the
location of disjoint objects relatively to one other.
Besides betweenness, ordering relations include:
left-of, right-of, in-front-of, above, below, behind,
etc. The science of anatomy has developed a whole
set of ordering relation terms to describe the
arrangement of anatomical parts in the human body:
superior, inferior, anterior, posterior, lateral,
medial, dorsal, ventral, rostral, proximal, distal, etc.
The FMA, for example, has an ‘orientation
network’ in which these kinds of relations are
represented [
Unfortunately, ordering relations between
spatially extended objects are difficult to formalize.
As [
To avoid problems that occur when describing
ordering relations between extended objects we will
choose a different approach: we will characterize
shape, extent, and spatial arrangement of
anatomical structures and their anatomical parts using
(point-like) anatomical landmarks [
Intuitively, anatomical landmarks are special
salient points on the surface of anatomical
structures or their anatomical parts [
LM1
LM2
LM3
LM4
LM6 LM5
R
LM7
A However not all salient points on the surface of a given anatomical structure are landmarks. Salient points are landmarks of anatomical structures of a given kind if and only if: 1. They exist as parts of every anatomical structure of that kind; 2. They are critical for the normal function of all anatomical structures of that kind.
Thus the salient points LM1-LM6 in Figure 3 are anatomical landmarks of temporal bones of normal human TMJs, since (a) they exist as parts of every temporal bones of a normal human TMJ and (b) they are important for the function of a human TMJ as a whole. Consequently, independently of the normal variations between the actual shape of temporal bones in different human beings, all normal temporal bones will have the landmarks LM1-LM7 as depicted in Figure 3.
Although normal temporal bones in human TMJs will have the landmarks LM1-LM7, the particular metric properties like the actual height of the maximum, the actual depth of the minimum, as well as their actual distance, will vary from individual to individual.
Consider the landmarks of the temporal bone depicted in Figure 3. Rather than quantitatively characterizing shape differences in terms of coordinate differences among the landmarks, we can characterize the shape differences qualitatively by specifying qualitative distance relations between those landmarks. Consider, for example, the anatomical landmarks LM1 and LM3. In Figure 3 the coordinate difference along the anterior (horizontal) axis is smaller than the coordinate difference 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 their temporal bones (assuming a certain degree of anatomical normality), we can classify TMJs according to qualitative coordinate differences between their landmarks. There are many ways of doing this. Here we only discuss some examples to demonstrate the power of the qualitative methodology. In particular we focus on the landmarks LM1, LM3, and LM5.
Given a coordinate system3 existing coordinate 3We do not need the coordinate system for measurement. We only use it to distinguish coordinate differences in anterior (horizontal) direction (δh) from coordinate differences in rostral (vertical) direction (δv). differences between LM1 and LM3 along the anterior axis (δa13) and along the rostral axis (δr31) can be used to distinguish the following cases: δa13 = δr31, δa13 < δr31, and δa13 > δr31. Here δa13 = δr31 means that δa13 is as large as δr31, δa13 < δr31 means that δa13 is smaller than δr31, and δa13 > δr31 means that δa13 is larger than δr31. Notice that this classification is jointly exhaustive and pairwise disjoint. That is, for any possible constellation of the anatomical landmarks LM1 and LM3 exactly one of those relations holds. In Figure 3 the rostral coordinate difference between LM1 and LM3 is larger than the anterior coordinate difference between LM1 and LM3, i.e., δa13 < δr31. Of course we can in addition classify the anterior and rostral coordinate differences between the landmarks LM3 and LM5 in the same way. If we take both classifications together then the following nine combinations are combinatorially possible:
R ∈ δ{a=31, <R,δ>r}31 = = = < < < > > > 1 2 3 4 5 6 7 8 9 δa53 R δr53 < > = < > = < > = Any possible constellation of LM1, LM2, and LM3 is characterized by exactly one column in this table. In Figure 3 we have δa13 < δr31 and δa35 > δr53. which corresponds to column 5 in the above table. Since this classification is exhaustive we now can analyze which of the nine possibilities are normal and which are pathological or which correlate with certain clinical symptoms. This analysis may show that distinguishing nine cases is insufficient to make the necessary distinction to distinguish normal anatomy form various kinds of pathologies. In this case we have three options: (a) take more landmarks into account; (b) distinguish more relations; (c) do both (a) and (b).
Consider option (b) instead of distinguishing three relations =, <, and > we could add two more relations: and interpreted as much smaller and much bigger respectively. Another way of distinguishing more relations would be to refine > by distinguishing twice as big, three times as big, etc. There are no limits to this method provided the resulting set of relations is jointly exhaustive and pairwise disjoint.
Notice that it might be more realistic to replace
the identity relation = by the relation ∼, were
δa ∼ δr means that δa is roughly as large as δr.
The exact definitions of the relations ∼, , and
are not trivial and their formalization is beyond
the scope of this paper. For discussions of existing
approaches see [
There exist a variety of approaches to qualitatively
represent angles between landmarks and to use
landmarks as origins for qualitative frames of
references. For example, the landmark ‘LM’ in
Figure 4(a) could serve as the origin of the qualitative
frame of reference in Figure 4(b). We then could
specify the location of anatomical landmarks of
the heart within this frame of reference.
Most of the approaches to qualitative orientation
and directions also incorporate qualitative
distance relations like close, near, far, etc. (where
close, near, and far roughly correspond to the
relations ∼, <, and – see for example, [
(a)
LM left near left near right near right far front, far front near close back near back far (b)
There are many ways to represent approximate
location in qualitative frames of references. (See, for
example [
In our mereotopological framework we can represent the topological relations between the intervals formed by the anatomical landmarks of Joe’s temporal bone as: Interval L1L2 is constantly externally connected to interval L2L3, interval L2L3 is constantly externally connected to interval L3L4, and so on.
LM3 LM4 LM5 LM6 LM7 LM3 LM4 LM5 LM6 LM7 LM2
LM2 R
A
R
A
Consider Figures 5(a) and (b) which depict the relative location of Joe’s articular disc with respect to his temporal bone at times t1 and t2 respectively. Figure 5(a) corresponds to Figure 1(a) and both show Joe’s TMJ in the jaw closed position. Similarly, Figure 5(b) corresponds to Figure 1(b) and both show Joe’s TMJ in the jaw open position. On the bottom of both images in Figure 5 the projection of Joe’s articular disc onto the boundary of his temporal bone is depicted. From this point on, we will write Prj(D, t) to refer the interval that is the projection of Joe’s articular disc on the boundary of his temporal bone in a sagittal section through the middle of his condyle at time t.
The interval Prj(D, t) stands in mereotopological
relationships to the intervals bounded by the
landmarks LM1-LM7. For example, at time t1 the
projection of Joe’s articular disc completely covers
the interval L3L4, i.e., COV(P rj(D, t1), L3L4, t1).
In other words the interval L3L4 is a part
of the projection of Joe’s articular disc, i.e.,
PartOf(L3L4, Prj(D, t1), t1). Notice that at
time t2 the projection of Joe’s articular disc
and the interval L3L4 are disconnected, i.e.,
DC(L3L4, Prj(D, t2), t2).4
Thus at every time t we can specify the location
of Joe’s articular disc with respect to the
landmarks of his temporal bone in terms of the
rela4For details of the exact definitions of the relations
between the intervals see [
L1L2 DC DC
L2L3 EC DC
L3L4 COV DC
L4L5 PO PO
L5L6 DC PO
L6L7 DC DC The first row reads as DC(Prj(D, t1), L1L2, t1), EC(Prj(D, t1), L2L3, t1), . . . and similarly for the second row.
Consider the images shown in Figures 6(a) and (b) which depict the relative location of Joe’s condyle with respect to his temporal bone at times t1 and t2 respectively. Figure 6(a) corresponds to Figure 1(a) and Figure 6(b) corresponds to Figure 1(b). In the same way we projected Joe’s disc onto the boundary of his temporal bone to identify an interval that can be related to the intervals bounded by the landmarks LM1-LM7, we can project the head of his condyle onto the boundary of his temporal bone as indicated by the dotted lines in Figures 6 (a) and (b).
LM3 LM4 LM5 LM6 LM7 LM3 LM4 LM5 LM6 LM7 LM2 LM1 LM2
LM1 (a)
R
A (b)
R
A As in the case of Joe’s disc, at every time t we can specify the location of the head of Joe’s condyle with respect to the landmarks of his temporal bone it terms of the relations which hold at time t between the projection the head of the condyle at t and the intervals bounded by the landmarks. The spatial relations at time t1 and t2 can be summarized as: If we use C to denote the head of Joe’s condyle then the first row reads as DC(Prj(C, t1), L1L2, t1), EC(Prj(C, t1), L2L3, t1), . . . , and similarly for the second row. Notice that the table with the relations of Joe’s articular disc corresponds nicely to the table with the relations of the head of Joe’s condyle, i.e., the articular disc is at both times between the head of the condyle and the temporal bone.
Clearly, for every possible location of an articular disc in a TMJ with respect to the temporal bone of this TMJ there is a unique sequence of relations similar to those in the table of Joe’s disc. Similarly, for every possible location of the head of a condyle in a TMJ with respect to the temporal bone of this TMJ there is a unique sequence of relations similar to those in the table of Joe’s condyle. Moreover, since we have, (i) the same anatomical landmarks on the temporal bones of every normal TMJ and, (ii) there are only a finite number of mereotopological relations that can hold between two intervals, we can therefore, compose two finite tables: one table in which each row corresponds to one anatomically possible location of some articular disc with respect to the corresponding temporal bone; a second table in which each row corresponds to one anatomically possible location of the head of some condyle with respect to the corresponding temporal bone.5 Both tables together contain all possible combinations of locations of the head of a condyle and an articular disc with respect to the landmarks of a temporal bone in any possible TMJ. Some of these combinations we can classify as normal (among these are the two tables above) others are pathological and again others will be anatomically impossible and thus can be ruled out.
The purpose of this paper is to show that there can be obtained, by following the methodology we have presented here, a series of well understood qualitative formalisms which can be used to create a formal representation of canonical anatomy. This is accomplished by incorporating into the representation, using the qualitative methods of analysis we describe in this paper, information about, a) the mereological (parthood) relationships of anatomical structures, b) the topology (e.g., connectedness) of anatomical structures, and c) the shape of anatomical parts and the spatial arrangement of anatomical structures.
The five cornerstones of the proposed
methodology are:
1. The grounding of the formalization of canonical
anatomy in mereotopology (rather than
mereology alone);
5For formal details of how to construct the tables
see [
This methodology permits, in principle, the exhaustive qualitative characterization of all anatomically possible instantiations of anatomical structures. These then can be classified as normal or pathological and correlated with other clinical findings.
The discussion in this paper exclusively focused
on relations between particulars (Joe Doe’s TMJ).
It is well known that anatomical ontologies are
mostly about relations between universals or
classes [
Thomas Bittner, State University of New York, Department of Philosophy, 135 Park Hall, Buffalo (NY), 14260, USA
A survey of image registration