Bioinformatics Advance Access originally published online on April 26, 2007
Bioinformatics 2007 23(13):1674-1682; doi:10.1093/bioinformatics/btm155
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The qualitative and time-dependent character of spatial relations in biomedical ontologies
1Departments of Philosophy and Geography, 2Departments of Oral Biology and Oral Diagnostic Sciences, School of Dental Medicine and 3New York State Center of Excellence in Bioinformatics and Life Sciences, University at Buffalo, USA
*To whom correspondence should be addressed.
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ABSTRACT |
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Motivation: The formal representation of mereological aspects of canonical anatomy (parthood relations) is relatively well understood. The formal representation of other aspects of canonical anatomy, such as connectedness and adjacency relations between anatomical parts, their shape and size as well as the spatial arrangement of anatomical parts within larger anatomical structures are, however, much less well understood and represented in existing computational anatomical and bio-medical ontologies only insufficiently.
Results: In this article, we provide a methodology of how to incorporate this kind of information into anatomical and bio-medical ontologies by applying techniques of representing qualitative spatial information from Artificial Intelligence. In particular, we focus on how to explicitly take into account the qualitative and time-dependent character of these relations. As a running example, we use the human temporomandibular joint (TMJ).
Availability: Using the presented methodology, a formal ontology was developed which is accessible on http://www.ifomis.org/bfo/fol. This ontology may help to improve the logical and ontological rigor of bio-medical ontologies such as the OBO relation ontology.
Contact: bittner3{at}buffalo.edu
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1 INTRODUCTION |
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Anatomical ontologies are formal representations of facts about the major parts of canonical anatomical structures, the qualitative shapes of those parts and qualitative relations between them (Donnelly et al., 2006; Mejino and Rosse, 2004; Rosse et al., 1998; Smith et al., 2005).
The formal representation of mereological aspects of canonical anatomy (parthood relations) is relatively well understood (Donnelly et al., 2006; Smith and Rosse, 2004), and has been implemented in computational medical ontologies like the FMA (Rosse and Mejino, 2003), GALEN (Rogers and Rector, 2000) and SNOMED (Spackman et al., 1997). On the other hand, the formal representation of other aspects of canonical anatomy like containment, connectedness and adjacency relations, shape and size properties and the spatial arrangement of anatomical parts within larger anatomical structures are less well understood and are represented only insufficiently in existing computational anatomical ontologies (Bittner and Donnelly, 2007b,c). This may result in errors where relations such as external connectedness are confused with adjacency relations (see Section 2.3). In this article, we propose a methodology of how to improve the representation of this kind of information in anatomical ontologies.
Following Rosse et al. (1998), 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 macroscopic 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 spatial relations need to hold between those parts. For example, some parts are connected to others, some parts are adjacent to others, some parts (like articular discs) need to be between other parts (like the bones in synovial joints), etc.
In this article, we give an overview of the most important qualitative spatial relations. We also demonstrate how the changes of spatial 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 or attached 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 (Brachman and Levesque, 1985; Weld and de Kleer, 1990). Cohn and Hazarika, (2001) stress that the essence of qualitative representations is to find ways to represent continuous properties of the world by discrete systems of symbols. As Forbus, (1984) points out, one can always quantize something continuous, but not all quantizations are equally useful because the distinctions made by a quantization must be relevant for the kind of reasoning performed. This is where formal ontology comes into play (Smith and Brogaard, 2002). It will be an important aspect of this article to show how to discretize continuous domains in such a way that ontologically significant properties are preserved.
We now discuss how to build such qualitative representations of canonical anatomical structures and use the human temporomandibular joint (TMJ) as a running example. Our discussion is purposely informal and focuses on the methodological aspects which, we believe, will provide the foundations for the next generation of bio-medical ontologies. Bittner and Donnelly (2007b) present a formal ontology that rigorously formalizes many of the ideas presented here informally.
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2 PARTHOOD, CONNECTEDNESS AND ADJACENCY RELATIONS |
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 TOPABSTRACT1 INTRODUCTION 2 PARTHOOD, CONNECTEDNESS AND... 3 PERMANENT PARTHOOD,...4 QUALITATIVE ORDERING RELATIONS...5 LANDMARKS6 APPROXIMATE LOCATION IN...7 DISCUSSION8 CONCLUSIONSREFERENCES |
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We start by identifying and classifying the macroscopic canonical parts of the TMJ and establish canonical mereotopological (parthood and connectedness) relations as well as adjacency relations between them.
2.1 Parthood relations
At the most basic level of the study of the canonical structure of TMJ, we consider its anatomical parts at the macroscopic level of granularity. That is, we consider salient singly connected parts of non-negligible size (thus cells and molecules are parts of anatomical structures but not anatomical parts of macroscopic scale).1 At this macroscopic anatomical level of granularity, we distinguish two kinds of anatomical parts: material parts and cavities.2 The material anatomical parts of the TMJ at the macroscopic anatomical level of granularity according to Laskin et al. (2006) are depicted in Figure 1, which shows, in a sagittal section through the middle of the condyle, a TMJ in closed (a) and open (b) jaw position: temporal bone (1), head of the condyle (2), articular disc (3), posterior attachment (4), lateral pterygoid muscle (5). Immaterial anatomical parts (cavities) include the superior and inferior synovial cavities, which are depicted as white spaces above and below the articular disc and the posterior attachment.
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A clear understanding of the number and kinds of canonical parts of an anatomical structure is 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 article, we refer to individual anatomical structures and their 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 time-dependent relation since anatomical 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.
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.
2.2 Connectedness relations
The ternary relation of connectedness holds between two objects x and y at a time instant t if and only if the distance between x and y is zero at t (where distance between extended objects is here understood as the greatest lower bound of the distance between any point of the first object and any point of the second object). 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 (Varzi, 1996).
Object x is singly connected at time t if and only if every two parts of x that jointly sum up to x at t are connected at t. Consider Joe's condyle at time t depicted partly in Figure 2a. Every two parts of Joe's condyle that sum up to Joe's condyle at t (e.g. the head of Joe's condyle and the rest of his condyle) are connected at t. According to the FMA, all normal human organs are singly connected wholes at all times at which they exist. (Unless they are injured or pathological.) By contrast, consider the bikini of Joe's wife. This object (like any other bikini) has two parts, the top and the bottom, which jointly sum up to the whole bikini but are not connected. Hence, the bikini of Joe's wife is not singly connected.
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Object x is a salient singly connected part of an anatomical structure if and only if at all times at which x exists, x is singly connected and a significant portion of x's boundary is of the bona fide sort. Bona fide boundaries are boundaries of material objects which coincide with discontinuities in the underlying reality. The exterior boundary of Joe's articular disc is a bona fide boundary. By contrast, the boundary between the head of Joe's condyle and the reminder of his condyle is a fiat boundary. Fiat boundaries are the result of human demarcations and do not correspond to discontinuities in the underlying reality. [see Smith (2001) for details.] Consider Joe's condyle. It has a fiat boundary which separates it from the reminder of Joe's mandible, but a significant portion of Joe's condyle has a bona fide boundary, i.e. only negligible parts of the boundary of Joe's condyle are of the fiat sort.
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. For example the head of Joe's condyle and the reminder of his condyle are externally connected. Externally connected objects share parts of their boundaries but no parts of their interiors.
Consider Figure 2. Every salient singly connected material macroscopic anatomical part of Joe's TMJs in Figure 1a and b is represented by the corresponding labels of nodes in the graphs in Figure 2. Nodes representing the superior and inferior synovial cavities are labeled ‘ssc’ and ‘isc’, respectively. The solid edges in the graphs in Figure 1a and b represent external connectedness relations which hold between the superior and inferior synovial cavities and parts of Joe's TMJ that bound the respective cavity as depicted in Figure 1a and b. For example, some parts of the boundary of Joe's superior synovial cavity coincide with parts of the (bona fide) boundary of his temporal bone. Since both objects share parts of their boundaries (and nothing else) the relation of external connectedness holds. [see Smith and Varzi (2000) for details on the relationships between bona fide boundaries of material objects and the boundaries of their cavities.]
Objects x and y are disconnected or (topologically) separated at time t if and only if x and y are not connected at t. Consider Figure 1a and b. Since, in the normal condition, the articular disc is between the temporal bone and the head of the condyle, both the temporal bone and the head of the condyle are disconnected. Similarly, the posterior attachment and the lateral pterygoid muscle are disconnected since the head of the condyle is between them. Following Smith and Varzi (2000), we will argue in Section 2.3 that for example the head of Joe's condyle and his lateral pterygoid muscle are adjacent and very close but nevertheless topologically separated or disconnected.
We introduce connectedness, separatedness and singly connectedness as time-dependent relations and properties since anatomical structures can be singly connected at some times and non-singly connected at other times. For example, Joe's condyle is singly connected at t and in particular the head of Joe's condyle is externally connected to the reminder of his condyle at time t. At time t2, Joe gets into a fist fight and Joe's condyle gets broken along the boundary between the head and the reminder of the condyle during the course of this fight. At t2, the head of Joe's condyle and the reminder of his condyle are separated (or disconnected). Thus Joe's condyle is not singly connected at t2. At time t3 after surgery and a successful healing process, the head of Joe's condyle and the reminder of his condyle are connected again and thus the condyle as a whole is singly connected at t3.
2.3 Adjacency relations
The relation adjacent-to holds among the macroscopic material parts of Joe's TMJ that surround the cavities. For example, the head of Joe's condyle is adjacent to his lateral pterygoid muscle. These two non-overlapping objects share no boundary parts and thus are not (externally) connected in the standard topological sense. Nonetheless, certain parts of the head of Joe's condyle and his lateral pterygoid muscle are adjacent or in close contact.
More precisely, object x is adjacent to object y at time t if and only if x and y are disconnected at t and there is a sphere-shaped region z that is negligible in size with respect to x and y and z overlaps the regions occupied by x and y at t.
Thus, Joe's condyle and his lateral pterygoid muscle are adjacent at t in the sense that at this time there is a sphere-shaped region that overlaps the region of the head of Joe's condyle and the region of his lateral pterygoid muscle, and this region is negligible in size with respect to both: the region of Joe's condyle and the region of his lateral pterygoid muscle. Similarly, Joe's condyle is adjacent to his posterior attachment, his articular disc is adjacent to both his temporal bone, and to the head of his condyle. In the graphs in Figure 2, the adjacency relation is represented by dashed edges.
We introduce adjacency as a time-dependent relation since anatomical structures can be adjacent at some times and not adjacent at other times. For example, at time t Joe's articular disc is adjacent to his lateral pterygoid muscle. At time t2, after the fight, Joe's articular disc may be dislocated and detached from (not adjacent to) his lateral pterygoid muscle.
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3 PERMANENT PARTHOOD, CONNECTEDNESS AND ADJACENCY RELATIONS |
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Parthood, connectedness and adjacency relations are time dependent. This means that a given relation may hold between two objects at one time but may fail to hold between the same two objects at another time. Notice, however, that in canonical anatomy parthood, connectedness and adjacency relations are permanent. To see this, consider Figures 2a, b and 3a. The nodes in the graphs represent permanent salient singly connected parts of Joe's TMJ. At all times at which the TMJ as a whole exists, the condyle (2) is a proper part of it. Similarly the temporal bone (1), the articular disc (3), the posterior attachment (4), the lateral pterygoid muscle (5), as well as the inferior and superior synovial cavities, are permanent proper parts of the TMJ.
Similarly, the connectedness and adjacency relations between the macroscopic parts of a TMJ are normally—if no injury occurs—permanent relations. The solid edges in the graphs in Figure 2a and b represent permanent external connectedness relations between major macroscopic parts of Joe's TMJ. The dashed edges in the graphs in Figure 2a and b and all edges in the graph in Figure 3a represent permanent adjacency relations. Joe's temporal bone is permanently externally connected to his SSC, i.e. at all times at which Joe's temporal bone or his SSC exist, the former is externally connected to the latter (and vice versa). Similarly, Joe's temporal bone is permanently adjacent to his articular disc, i.e. at all times at which Joe's temporal bone or his synovial disc exist, the former it is adjacent to the latter (and vice versa).
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We define: relation R permanently holds between objects x and y if and only if the relation Rt holds between x and y at all times t at which x exists or y exists. R and Rt are meta-variables. Rt can be any of the time-dependent relations discussed above and R can be any corresponding permanent relations. For example, object x is a permanent part of object y if and only if whenever x exists or y exists, x is a part of y. Object x is a permanently externally connected to object y if and only if whenever x exists or y exists, x is externally connected to y. Similarly for connected-to, adjacent-to, overlaps, etc.
Now compare the permanent adjacency relation between Joe's temporal bone and his lateral pterygoid muscle and the permanent adjacency relation between Joe's articular disc and his temporal bone. The permanent adjacency relation between Joe's temporal bone and his lateral pterygoid muscle is different (with respect to time) from the permanent adjacency relation that holds between his articular disc and his temporal bone: Joe's temporal bone and his lateral pterygoid muscle are strictly permanently adjacent or attached in the sense that at all times at which they are in the adjacency relation, the same parts of the temporal bone and the lateral pterygoid muscle are in the adjacency relation.
Joe's articular disc, on the other hand, has the capability to slide along the temporal bone. That is, at all times Joe's articular disc is adjacent to his temporal bone but at different times his disc is adjacent to different parts of the temporal bone. Joe's articular disc is variabley permanently adjacent to his temporal bone. Consider Figure 1a and b. At time ta, Joe's articular disc is adjacent to his fossa (a fiat part of Joe's temporal bone). At time tb, Joe's articular disc is adjacent to the articular eminence (another fiat part of the temporal bone).
It is important to make explicit that the temporal character of the (permanent) adjacency relation between the articular disc and the temporal bone is different from the temporal character of the (permanent) adjacency relation between the temporal bone and the lateral pterygoid muscle.
Object x is strictly permanently adjacent (i.e. attached) to object y if and only if (i) x and y are permanently adjacent, (ii) if 
is a permanent macroscopic proper part (a permanent proper part of non-negligible size) of x, (iii) if 
is a permanent macroscopic proper part of y and (iv) 
and 
are adjacent at some time, then 
and 
are permanently adjacent. Joe's lateral pterygoid muscle is strictly permanently adjacent (i.e. attached) to his temporal bone.
Object x is non-strictly permanently adjacent to object y if and only if x is permanently adjacent to y but not strictly permanently adjacent to y. Joe's articular disc is non-strictly permanently adjacent to his temporal bone. In the graph in Figure 3c strict permanent adjacency (attachment) is represented by solid edges and non-strict permanent adjacency is represented by dotted edges between the respective nodes.
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4 QUALITATIVE ORDERING RELATIONS BETWEEN EXTENDED OBJECTS |
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TOPABSTRACT1 INTRODUCTION2 PARTHOOD, CONNECTEDNESS AND...3 PERMANENT PARTHOOD,...4 QUALITATIVE ORDERING RELATIONS...5 LANDMARKS6 APPROXIMATE LOCATION IN...7 DISCUSSION8 CONCLUSIONSREFERENCES |
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Parthood, connectedness and adjacency relations alone are not powerful enough to sufficiently characterize the important properties of TMJs. Consider the graphs in Figures 2a, b and 3a, which are representations of the permanent mereotopological and adjacency relations between the salient singly connected parts of the TMJs depicted in Figures 1a, b and 3b. The fact that the TMJs depicted in the three figures have the same graph-theoretic representations shows that in terms of parthood, connectedness and adjacency relations we cannot distinguish the TMJs in Figures 1a, b and 3b. This is because all three TMJs are equivalent, i.e. indistinguishable, with respect to their parthood, connectedness and adjacency structure.
Obviously it is critical to distinguish the TMJ in Figure 3b from the TMJs in Figure 1a and 1b. It is the purpose of the articular disc in a TMJ to be between the head of the condyle and temporal bone at all times. If we take the ordering relation of betweenness into account, then the TMJs in Figure 1a and b can be distinguished from the clearly pathological TMJ in Figure 3b 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 (Hernandez, 1994). 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, has an ‘orientation network’ in which these kinds of relations are represented (Mejino and Rosse, 2004; Rosse and Mejino, 2003).
Unfortunately, ordering relations between spatially extended objects are difficult to formalize. As Donnelly (2001) points out in her treatment of relation of betweenness: ‘The problem with trying to characterize the betweenness relation on extended objects is that we typically use the betweenness relation only on objects that have fairly uniform shapes and are nearly the same size. It is unclear whether or not the betweenness relation should hold in certain cases involving irregularly shaped objects and differently sized objects’.
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5 LANDMARKS |
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To avoid the problems that occur when characterizing 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 (Brown, 1992) and qualitative ordering and distance relations between the landmarks.
Intuitively, anatomical landmarks are ontologically salient point-like parts of negligible size (points for short) on the surface or within anatomical structures or their anatomical parts (Brown, 1992). Consider Figure 4 which shows, in a sagittal section through Joe's temporal bone at the level of the middle of the condyle. Salient points on the inferior surface of the temporal bone are local minima (LM3, LM7), local maxima (LM1, LM5) as well as points at which changes from convexity to concavity occur (LM2, LM4, LM6).
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Notice that not all salient points on the surface or within a given anatomical structure are landmarks. Anatomical landmarks are ontologically salient points of certain spatial characteristic
(local minimum, local maximum, ...) that exist in all anatomical structures of a given kind 
. More precisely, if P is a -landmark for anatomical structures of kind then: (1) P is a part of negligible size (a point-like part) of characteristic of an anatomical structure of kind ; (2) every anatomical structure of kind has a corresponding point-like part 
of the same characteristic ' and (3) corresponding point-like parts of characteristic are critical for the normal function of all anatomical structures of kind . The salient points LM1–LM7 in Figure 4 are anatomical landmarks of temporal bones of normal human TMJs, since (a) points of the same characteristic exist as parts of (sagittal sections of) every temporal bone of a normal human TMJ and (b) in every particular TMJ these points are important for the function of that TMJ as a whole. Consequently, independently of the normal variations between the actual shape of temporal bones in different human beings, all (sagittal sections of) normal temporal bones will have the landmarks LM1–LM7 as depicted in Figure 4.
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 from each other, will vary from individual to individual. Consider the landmarks of the temporal bone depicted in Figure 4. 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, the anatomical landmarks LM1 and LM3. In Figure 4, 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 differences between LM1 and LM3 along the anterior axis (
) and along the rostral axis (
) can be used to distinguish the following cases: 
, 
and 
. Here, 
means that 
is as large as 
, 
means that 
is smaller than 
and 
means that 
is larger than 
. 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 4, the rostral coordinate difference between LM1 and LM3 is larger than the anterior coordinate difference between LM1 and LM3, i.e. 
.
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 nine combinations listed in Table 1 are combinatorially possible. Any possible constellation of LM1, LM2 and LM3 is characterized by exactly one column in this table. In Figure 4, we have 
and 
, which corresponds to column 5 of Table 1.
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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 differentiation between normal anatomy and various kinds of pathologies. In this case, we have three options: (a) take more landmarks into account; (b) distinguish more relations and (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 ‘between twice as big and three times as big’, ‘between three times and four 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 
means that 
is roughly as large as 
. For existing approaches to formalize qualitative size and distance relations like , << and >> (Bittner and Donnelly, 2006, 2007b; Dague, 1993; Raiman, 1991).
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6 APPROXIMATE LOCATION IN QUALITATIVE FRAMES OF REFERENCE |
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There are many ways to represent approximate location in qualitative frames of references (Hernandez, 1994). Here, we discuss a specific technique which is useful in the context of our TMJ example because it enables us to capture the dynamic character of the joint in a qualitative manner.
Consider the inferior boundary of the sagittal section of Joe's temporal bone as depicted in Figure 4. Topologically, the boundary is a 1D curve. Since the landmarks LM1–LM7 are points on this curve, each landmark is a boundary of at least one interval (a singly connected part of the underlying curve). For example, in Figure 4 the landmarks LM2 and LM3 bound the interval which is formed by the part of the curve between them. We use the landmarks that bound a given interval to refer to this interval. For example, we write 
to refer to the interval bounded by LM2 and LM3 in Figure 4.
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 
is permanently externally connected to interval 
, interval 
is permanently externally connected to interval 
, and so on.
Consider Figure 5a and b which depict the relative location of Joe's articular disc with respect to his temporal bone at times ta and tb, respectively. Figure 5a corresponds to Figure 1a and both show Joe's TMJ in the jaw closed position. Similarly, Figure 5b corresponds to Figure 1b 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 
to refer to the interval that is the projection of Joe's articular disc on the inferior boundary of his temporal bone in a sagittal section through the middle of his condyle at time t.
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The interval
stands in mereotopological relationships to the intervals bounded by the landmarks LM1–LM7. For example, at time ta the projection of Joe's articular disc completely covers the interval 
, i.e. 
. In other words the interval 
is a part of the projection of Joe's articular disc, i.e. 
. Notice that at time tb the projection of Joe's articular disc and the interval 
are disconnected, i.e. 
. [For details of the exact definitions of the relations between the intervals see (Allen, 1983; Bittner, 2002)].
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 relations which hold at time t between the projection of the articular disc at t and the intervals bounded by the landmarks. These mereotopological relations at time ta and tb are summarized in the first two rows of Table 2. The first row of the table reads as 
, 
, ... and similarly for the second row.
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Consider the images shown in Figure 6a and b which depict the relative location of the head of Joe's condyle with respect to his temporal bone at times ta and tb, respectively. Figure 6a corresponds to Figure 1a and Figure 6b corresponds to Figure 1b. 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 inferior boundary of his temporal bone as indicated by the dotted lines in Figure 6a and b.
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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 in terms of the relations which hold at time t between the projection of the head of the condyle at t and the intervals bounded by the landmarks. The spatial relations at time ta and tb are summarized in rows 3 and 4 of Table 2. In this table, we use C to denote the head of Joe's condyle. The third row reads as
, 
, ..., and similarly for the fourth row. Notice that, at corresponding times, the rows with the relations of Joe's articular disc corresponds nicely to the rows 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. |
7 DISCUSSION |
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The cornerstones of the methodology presented in this article are: (1) the strict distinction of time-dependent and time-independent (i.e. permanent) relations; (2) the grounding of the formalization of canonical anatomy in permanent mereotopological and adjacency relations and (3) the representation of qualitative distinctions that reflect the ontologically significant aspects of spatial arrangement of the parts of anatomical structures in terms of landmarks and sets of jointly exhaustive and pairwise disjoint sets of relations between anatomical parts and anatomical landmarks.
The representation of permanent mereotopological and adjacency relations between parts of anatomical structures yields graph structures similar to those in Figures 2a, b, and 3a. Those graphs (which are not always planar) represent structural properties that remain invariant under normal changes of the arrangement of the parts of the structure at hand.
The possibly changing spatial arrangement of anatomical parts is represented using qualitative frames of reference formed by anatomical landmarks. We discussed this in the context of the qualitative description of the location of Joe's articular disc and the head of his condyle with respect to landmarks of (the inferior boundary of a sagittal section of) his temporal bone (Table 2). In general, for every possible location of an articular disc in a TMJ with respect to the temporal bone there is a unique sequence of relations similar to those in Table 2. Similarly, for every possible location of the head of a condyle in a TMJ with respect to the temporal bone there is a unique sequence of relations similar to those in the Table 2.
Moreover, since (i) we have the same anatomical landmarks on the temporal bones of every normal TMJ, (ii) there are only finitely many intervals between those landmarks and (iii) 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. 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 them are those represented in Table 2) others are pathological and again others will be anatomically impossible and thus can be ruled out.
Notice that even though our example is rather specific, the proposed method is quite general. Most anatomical structures have parts that are connected or adjacent to another. Most parts have landmarks and landmarks stand in spatial relations to one another that can be used to define a frame of reference, which then can be used to represent ordering relations between the anatomical parts. Representation and reasoning aspects of such an approach were discussed extensively in Bittner (2002).
In the presentation of our methodology, we frequently mentioned qualitative size measures such as ‘roughly the same size-as’ and ‘negligible in size with respect to’. We have shown that qualitative size measures are important for the formalization of adjacency relations between disconnected but very close anatomical parts.
Qualitative size measures are also important for distinguishing parts of the same scale (Joe's articular disc is a same-scale-part of his TMJ) from granular parts — parts of negligible size — (a cell of Joe's articular disc is a granular part of his TMJ). This distinction is important because there are significant differences in the temporal properties of spatial relations between macroscopic and microscopic parts of the human body: most macroscopic parts of the human body are permanent parts of the human body. By contrast, many microscopic parts are non-permanent. For example, most of the the cells that are parts of Joe's lateral pterygoid muscle at some time fail to be permanent parts of his lateral pterygoid muscle (Bittner and Donnelly, 2007d). Thus, it is critical to include qualitative size relations into bio-medical ontologies.
The discussion in this article 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 (Smith and Rosse, 2004; Smith et al., 2005). However, it is also well known that relations between universals or classes are defined in terms of relations between particulars using well known definitorial patterns (Donnelly et al., 2006).
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8 CONCLUSIONS |
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The purpose of this article was to present a methodology designed to create qualitative representations of canonical anatomy. This is accomplished by incorporating into the representation (a) the mereotopology (parthood and connectedness structure) of anatomical structures, (b) adjacency relations between anatomical parts, (c) the shape of anatomical parts and (d) the spatial arrangement of parts in larger anatomical structures. The presented methodology permits, in principle, the exhaustive qualitative characterization of all anatomically possible instantiations of anatomical structures and their temporal behavior. These then can be classified as normal or pathological and correlated with other clinical findings.
The presentation of the methodology was purposely informal and needs to be complemented by a formal ontology. Bittner and Donnelly (2007a,b) present an axiomatic theory that rigorously formalizes many of the relations discussed here informally. This theory is intended to extend the Relation Ontology (Smith et al., 2005), which in turn serves as a foundational ontology for the other ontologies in the OBO Foundry. A computational representation of the theory is accessible [as a part of ‘Basic Formal Ontology’ (BFO)] via http://www.ifomis.org/bfo/fol. Some of the logical properties of the qualitative relations discussed in this article are listed in Table 3 (for fixed times). The table also shows which symbols are used in the computational representation (corresponding permanent relations have a ‘p’ as prefix) and gives the names of the modules in which the axioms, definitions and theorems that capture the logical properties of the respective relations can be found.
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Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Alex Bateman
1 We will define more precisely what we mean by ‘singly connected’ in Section 2.2. For a formal specification of the meaning of ‘negligible in size’ and ‘non-negligible in size’ see (Bittner and Donnelly, 2006, 2007b). 
2 We follow Mejino and Rosse (2004) in considering cavities of an organ as parts of an organ (Joe's superior and inferior synovial cavities are parts of his TMJ, Joe's heart chambers are parts of his heart, etc.). Notice, however, that this is not uncontroversial. For example Casati and Varzi (1994) argue against considering cavities as parts of their enclosing objects.
3 We do not need the coordinate system for measurement. We only use it to distinguish coordinate differences in anterior (horizontal) direction (
h) from coordinate differences in rostral (vertical) direction (v).
Received on December 14, 2006; revised on April 8, 2007; accepted on April 17, 2007
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