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A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies

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Title: A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies


1
A Formal Theory for Spatial Representation and
Reasoning in Biomedical Ontologies
  • Maureen Donnelly
  • Thomas Bittner

2
Outline
  1. A formal theory of inclusion relations among
    individuals (BIT)
  2. Defining inclusion relations on classes
  3. Properties of class relations
  4. Parthood and containment relations in the FMA and
    GALEN

3
I. A formal theory of inclusion relations among
individuals (BIT)
4
Inclusion Relations
  • By inclusion relations we mean mereological and
    location relations.
  • We introduce 3 mereological relations
  • part (P), proper part (PP), and overlap (O)
  • We introduce 2 location relations
  • located-in (Loc-In) (e.g. my heart is
    located-in my thoracic cavity)
  • partial coincidence (PCoin) (e.g. my esophagus
    partially coincides with my thoracic cavity)

5
Properties of Mereological Relations
  • Parthood (P) is
  • reflexive, antisymmetric, and transitive
  • Proper Parthood (PP) is
  • irreflexive, asymmetric, and transitive
  • Overlap (O) is
  • reflexive and symmetric

6
Properties of Location Relations
  • Loc-In is
  • reflexive and transitive
  • Loc-In(x, y) Pyz ? Loc-In(x, z)
  • Pxy Loc-In(y, z) ? Loc-In(x, z)
  • PCoin is
  • reflexive and symmetric

7
Inverse Relations
  • The inverse of a binary relation R is the
    relation R-1xy if and only if Ryx
  • Inverses of the mereological and location
    relations are included in BIT.
  • For example,
  • PP-1(my body, my hand)
  • Loc-In-1(my thoracic cavity, my heart)

8
II. Defining inclusion relations on classes
9
Why define spatial relations on classes?
  • Biomedical ontologies like the FMA and GALEN
    contain only assertions about classes (not
    assertions about individuals).
  • These assertions include many claims about
    parthood and containment relations among classes
  • Right Ventricle part_of Heart
  • Uterus contained_in Pelvic Cavity
  • A formal theory of inclusion relations on classes
    can help us analyze these kinds of assertions and
    find appropriate automated reasoning procedures
    for biomedical ontologies.

10
Classes and Instances
  • Inst is introduced as binary relation between an
    individual and a class, where Inst(x, A) is
    intended as
  • individual x is an instance of class A
  • Inst(my heart, Heart)

11
Three types of inclusion relations among classes
  • R1(A, B) ?x (Inst(x, A) ? ?y(Inst(y, B)
    Rxy))
  • (every A is stands in relation R to some B)
  • R2(A, B) ?y (Inst(y, B) ? ?x(Inst(x, A)
    Rxy))
  • (for each B there is some A that stands in
    relation R to it)
  • R12(A, B) R1(A, B) R2(A, B)
  • (every A stands in relation R to some B and for
    each B there is some A that stands in relation R
    to it)

12
Examples of different types of class relations
PP1, PP2, and PP12
  • PP1(A, B) ?x (Inst(x, A) ? ?y(Inst(y, B)
    PPxy))
  • (every A is a proper part of some B)
  • Example PP1(Uterus, Pelvis)
  • PP2(A, B) ?y (Inst(y, B) ? ?x(Inst(x, A)
    PPxy))
  • (every B has some A as a proper part)
  • Example PP2(Cell, Heart)

    (but NOT PP2(Uterus, Pelvis) and NOT
    PP1(Cell, Heart))
  • PP12(A, B) PP1(A, B) PP2(A, B)
  • (every A is a proper part of some B and every B
    has some A as a proper part)
  • Example PP12(Left Ventricle, Heart)

13
Examples of different types of class relations
Loc-In1, Loc-In2, and Loc-In12
  • Loc-In1(A, B) ?x (Inst(x, A) ? ?y(Inst(y, B)
    Loc-In(x,y)))
  • (every A is located in some B)
  • Example Loc-In1(Uterus, Pelvic Cavity)
  • Loc-In2(A, B) ?y (Inst(y, B) ? ?x(Inst(x, A)
    Loc-In(x,y)))
  • (every B has some A located in it)
  • Example Loc-In2(Urinary Bladder, Male Pelvic
    Cavity)
    (but NOT
    Loc-In2(Uterus, Pelvic Cavity) and NOT
    Loc-In1(Urinary Bladder, Male Pelvic Cavity))
  • Loc-In12(A, B) Loc-In1(A, B) Loc-In2(A, B)
  • (every A is located in some B and every B has
    some A located in it)
  • Example Loc-In12(Brain, Cranial Cavity)

14
III. Properties of class relations
15
Properties of relations among individuals vs.
properties of relations among classes
Among Individuals Among Classes Among Classes Among Classes
R is... R1 must also be...? R2 must also be...? R12 must also be...?
Reflexive Yes Yes Yes
Irreflexive No No No
Symmetric No No Yes
Asymmetric No No No
Antisymmetric No No No
Transitive Yes Yes Yes
16
Inverses of Class Relations
  • The inverse of R12 is (R-1)12.
  • But...
  • the inverse of R1 is (R-1)2 and
  • the inverse of R2 is (R-1)1.
  • Example the inverse of PP1 is (PP-1)2
  • PP1(Uterus, Pelvis) is equivalent to
  • (PP-1)2(Pelvis, Uterus)
  • and NOT equivalent to (PP-1)1(Pelvis, Uterus)

17
Some inferences supported by our theory
PP1(B, C) PP2(B, C) PP12(B, C) Loc-In1(B, C) Loc-In2(B, C) Loc-In12(B,C)
PP1(A, B) PP1(A, C) PP1(A, C) Loc-In1(A, C) Loc-In1(A, C)
PP2(A, B) PP2(A, C) PP2(A, C) Loc-In2(A, C) Loc-In2(A, C)
PP12(A, B) PP1(A, C) PP2(A, C) PP12(A, C) Loc-In1(A, C) Loc-In2(A, C) Loc-In12(A, C)
Loc-In1(A, B) Loc-In1(A, C) Loc-In1(A, C) Loc-In1(A, C) Loc-In1(A, C)
Loc-In2(A, B) Loc-In2(A, C) Loc-In2(A, C) Loc-In2(A, C) Loc-In2(A, C)
Loc-In12(A, B) Loc-In1(A, C) Loc-In2(A, C) Loc-In12(A, C) Loc-In1(A, C) Loc-In2(A, C) Loc-In12(A, C)
18
Some inferences supported by our theory
Is_a(C, A) Is_a(A, C) Is_a(C, B) Is_a(B, C)
PP1(A, B) PP1(C, B) PP1(A, C)
PP2(A, B) PP2(C, B) PP2(A, C)
PP12(A, B) PP1(C, B) PP2(C, B) PP2(A, C) PP1(A, C)
19
IV. Parthood and containment relations in the FMA
and GALEN
20
Class Parthood in the FMA
  • The FMA uses part_of as a class parthood
    relation.
  • has_part is used as the inverse of part_of

21
Examples of FMA assertions using part_of
the FMAs part_of BITCl relation
1a Female Pelvis part_of Body PP1
1b Male Pelvis part_of Body PP1
2 Cavity of Female Pelvis part_of Abdominal Cavity PP1
3a Urinary Bladder part_of Female Pelvis PP2
3b Urinary Bladder part_of Male Pelvis PP2
4 Cell part_of Tissue PP2
5 Right Ventricle part_of Heart PP12
6 Urinary Bladder part_of Body PP12
7 Nervous System part_of Body PP12
22
Class parthood in GALEN
  • GALEN uses isDivisionOf as one of its most
    general class parthood relations
  • isDivisionOf behaves in most (but not all) cases
    as a restricted version of PP1
  • GALEN has a correlated relation hasDivision which
    it designates as the inverse of isDivisionOf
  • But, hasDivision is not used as the inverse of
    isDivisionOf. Rather, it behaves in most cases as
    a restricted version of (PP-1)1 (which is the
    inverse of PP2, NOT the inverse of PP1).
  • GALEN usually (but not always) asserts both A
    isDivisionOf B and B hasDivision A when PP12(A,
    B) holds. (note that PP12(A, B) is equivalent to
    PP1(A, B) (PP-1)1(A, B).)

23
GALEN assertions using isDivisionOF and
hasDivision
GALENs isDivisionOf assertion BITCl relation GALENs hasDivision BITCl relation
Female Pelvic Cavity isDivisionOf Pelvic Part of Trunk PP1 none
Prostate Gland isDivisionOf Genito-Urinary System PP1 none
none Pelvic Part of Trunk hasDivision Hair (PP-1)1
LeftHeartVentricle isDivisionOf Heart PP12 Heart hasDivision LeftHeartVentricle (PP-1)12
Prostate Gland isDivisionOf Male Genito-Urinary System PP12 Male Genito-Urinary System hasDivision Prostate Gland (PP-1)12
Urinary Bladder isDivisionOf Genito-Urinary System PP12 none
Pericardium isDivisionOf Heart none Heart hasDivision Pericardium none
24
The FMAs containment relation
  • The FMAs uses contained_in as a class location
    relation
  • A contained_in B holds only when A is a class of
    material individuals and B is a class of
    immaterial individuals
  • contained_in is used (in most cases) as either a
    restricted version of Loc-In1, Loc-In2, or
    Loc-In12.
  • contains is used as the inverse of contained_in.

25
FMA assertions using contained_in
the FMAs contained_in BITCl relation
1 Right Ovary contained_in Abdominopelvic Cavity Loc-In1
2a Urinary Bladder contained_in Cavity of Female Pelvis Loc-In2
2b Urinary Bladder contained_in Cavity of Male Pelvis Loc-In2
3 Blood contained_in Cavity of Cardiac Chamber Loc-In2
4 Urinary Bladder contained_in Pelvic Cavity Loc-In12
5 Uterus contained_in Cavity of Female Pelvis Loc-In12
6 Prostate contained_in Cavity of Male Pelvis Loc-In12
7 Heart contained_in Middle Mediastinal Space Loc-In12
8 Blood contained_in Lumen of Cardiovascular System Loc-In12
9 Bolus of Food contained_in Lumen of Esophagus none
26
Class containment in GALEN
  • GALEN uses isContainedIn as one of its most
    general class containment relations
  • isContainedIn behaves in many (but not all)
    cases as a restricted version of Loc-In1
  • GALEN has a correlated relation Contains which it
    designates as the inverse of isContainedIn
  • But, Contains is not used as the inverse of
    isContainedIn. Rather, it behaves in most cases
    as a restricted version of (Loc-In-1)1 (which
    is the inverse of Loc-In2, NOT the inverse of
    Loc-In1).
  • GALEN usually (but not always) asserts both A
    isContaindIn B and B Contains A when Loc-In12(A,
    B) holds. (note that Loc-In12(A, B) is equivalent
    to Loc-In1(A, B) (Loc-In-1)1(A, B).)

27
GALAN assertions using isContainedIn and Contains
GALENs isContainedIn BITCl relation GALENs Contains BITCl relation
1 Ovarian Artery isContainedIn Pelvic Cavity Loc-In1 Pelvic Cavity Contains Ovarian Artery (Loc-In-1)2
2 Uterus isContainedIn Pelvic Cavity Loc-In1 none
3 none Venous Blood Contains Haemoglobin (Loc-In-1)1
4 none Male Pelvic Cavity Contains Urinary Bladder (Loc-In-1)1
5 Uterus isContainedIn Female Pelvic Cavity Loc-In12 Female Pelvic Cavity Contains Uterus (Loc-In-1)12
6 Mediastinum isContainedIn Thoracic Space Loc-In12 Thoracic Space Contains Mediastinum (Loc-In-1)12
7 Larynx isContainedIn Neck Loc-In12 Neck Contains Larynx (Loc-In-1)12
8 Lung isContainedIn Pleural Membrane none Pleural Membrane Contains Lung none
9 Tooth isContainedIn Tooth Socket none Tooth Socket Contains Tooth none
10 none Male Pelvic Cavity Contains Ovarian Artery none
28
Also in GALEN...
  • Vomitus Contains Carrot
  • Speech Contains Verbal Statement
  • Inappropriate Speech Contains Inappropriate
    Verbal Statement

29
  • Male Pelvic Cavity Contains Ovarian Artery
  • seems to be inferred from
  • Pelvic Cavity Contains Ovarian Artery
  • and
  • Male Pelvic Cavity Is_a
  • Pelvic Cavity

30
BITCl Inferences
Is_a(C, A) Is_a(A, C) Is_a(C, B) Is_a(B, C)
(Loc-In-1)1(B, A) (Loc-In-1)1(B, C) (Loc-In-1)1(C, A)
(Loc-In-1)2(B, A) (Loc-In-1)2(B, C) (Loc-In-1)2(C, A)
(Loc-In-1)12(B, A) (Loc-In-1)2(B, C) (Loc-In-1)1(B, C) (Loc-In-1)1(C, A) (Loc-In-1)2(C, A)
31
Conclusions
32
  • Relational terms do not have clear semantics in
    existing biomedical ontologies.
  • Possibilities for expanding the inference
    capabilities of biomedical ontologies are
    limited, in part because they do not explicitly
    distinguish R1, R2, and R12 relations.
  • Given the (limited) existing reasoning structures
    in the FMA and GALEN, certain kinds of anatomical
    information cannot be added to these ontologies
    (without generating false assertions).
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