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 • A formal theory of inclusion relations among individuals (BIT) • Defining inclusion relations on classes • Properties of class relations • 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_inPelvic 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

16. Inverses of Class Relations The inverse of R12is (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

18. Some inferences supported by our theory

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

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

24. 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.

25. FMA assertions using contained_in

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

28. Also in GALEN... • Vomitus Contains Carrot • Speech Contains Verbal Statement • Inappropriate Speech Contains Inappropriate Verbal Statement

29. Pelvic Cavity Contains (Loc-In-1)2 SubclassOf Is_a Ovarian Artery Male Pelvic Cavity Contains 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. BIT+Cl Inferences

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).