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A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies. Maureen Donnelly Thomas Bittner. Outline. A formal theory of inclusion relations among individuals (BIT) Defining inclusion relations on classes Properties of class relations - PowerPoint PPT Presentation
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A Formal Theory for Spatial Representation and Reasoning in
Biomedical OntologiesMaureen Donnelly
Thomas Bittner
Outline
I. A formal theory of inclusion relations among individuals (BIT)
II. Defining inclusion relations on classes
III. Properties of class relations
IV. Parthood and containment relations in the FMA and GALEN
I. A formal theory of inclusion relations among individuals (BIT)
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)
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
Properties of Location RelationsLoc-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
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)
II. Defining inclusion relations on classes
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 HeartUterus 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.
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)
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)
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)
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)
III. Properties of class relations
Properties of relations among individuals vs. properties of relations among classes
Among Individuals
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
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)
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)
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)
IV. Parthood and containment relations in the FMA and GALEN
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
Examples of FMA assertions using part_of
the FMA’s part_of BIT+Cl 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
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).)
GALEN assertions using isDivisionOF and hasDivision
GALEN’s isDivisionOf assertion BIT+Clrelation
GALEN’s hasDivision BIT+Clrelation
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
The FMA’s containment relation
• The FMA’s 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.
FMA assertions using contained_in
the FMA’s contained_in BIT+Cl 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
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).)
GALAN assertions using isContainedIn and Contains
GALEN’s isContainedIn BIT+Clrelation
GALEN’s Contains BIT+Clrelation
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
Also in GALEN...
• Vomitus Contains Carrot
• Speech Contains Verbal Statement
• Inappropriate Speech Contains Inappropriate Verbal Statement
Male Pelvic Cavity Contains Ovarian Artery
seems to be inferred from
Pelvic Cavity Contains Ovarian Artery
and
Male Pelvic Cavity Is_a Pelvic Cavity
Contains
Pelvic Cavity
Male Pelvic Cavity
Ovarian Artery
SubclassOfIs_a
Contains(Loc-In-1)2
BIT+Cl 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)
Conclusions
• 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).