A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies Maureen 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 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

Inverse Relations The inverse of a binary relation R is the relation R -1 xy 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 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.

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 R 1 (A, B) =:  x (Inst(x, A)   y(Inst(y, B) & Rxy)) (every A is stands in relation R to some B) R 2 (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) R 12 (A, B) =: R 1 (A, B) & R 2 (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: PP 1, PP 2, and PP 12 PP 1 (A, B) =:  x (Inst(x, A)   y(Inst(y, B) & PPxy)) (every A is a proper part of some B) Example: PP 1 (Uterus, Pelvis) PP 2 (A, B) =:  y (Inst(y, B)   x(Inst(x, A) & PPxy)) (every B has some A as a proper part) Example: PP 2 (Cell, Heart) (but NOT: PP 2 (Uterus, Pelvis) and NOT: PP 1 (Cell, Heart)) PP 12 (A, B) =: PP 1 (A, B) & PP 2 (A, B) (every A is a proper part of some B and every B has some A as a proper part) Example: PP 12 (Left Ventricle, Heart)

Examples of different types of class relations: Loc-In 1, Loc-In 2, and Loc-In 12 Loc-In 1 (A, B) =:  x (Inst(x, A)   y(Inst(y, B) & Loc-In(x,y))) (every A is located in some B) Example: Loc-In 1 (Uterus, Pelvic Cavity) Loc-In 2 (A, B) =:  y (Inst(y, B)   x(Inst(x, A) & Loc-In(x,y))) (every B has some A located in it) Example: Loc-In 2 (Urinary Bladder, Male Pelvic Cavity) (but NOT: Loc-In 2 (Uterus, Pelvic Cavity) and NOT: Loc-In 1 (Urinary Bladder, Male Pelvic Cavity)) Loc-In 12 (A, B) =: Loc-In 1 (A, B) & Loc-In 2 (A, B) (every A is located in some B and every B has some A located in it) Example: Loc-In 12 (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...R 1 must also be...? R 2 must also be...?R 12 must also be...? ReflexiveYes IrreflexiveNo SymmetricNo Yes AsymmetricNo AntisymmetricNo TransitiveYes

Inverses of Class Relations The inverse of R 12 is (R -1 ) 12. But... the inverse of R 1 is (R -1 ) 2 and the inverse of R 2 is (R -1 ) 1. Example: the inverse of PP 1 is (PP -1 ) 2 PP 1 (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 PP 1 (B, C)PP 2 (B, C)PP 12 (B, C)Loc-In 1 (B, C)Loc-In 2 (B, C)Loc-In 12 (B,C) PP 1 (A, B)PP 1 (A, C) Loc-In 1 (A, C) PP 2 (A, B)PP 2 (A, C) Loc-In 2 (A, C) PP 12 (A, B)PP 1 (A, C)PP 2 (A, C)PP 12 (A, C)Loc-In 1 (A, C)Loc-In 2 (A, C)Loc-In 12 (A, C) Loc-In 1 (A, B)Loc-In 1 (A, C) Loc-In 2 (A, B)Loc-In 2 (A, C) Loc-In 12 (A, B)Loc-In 1 (A, C)Loc-In 2 (A, C)Loc-In 12 (A, C)Loc-In 1 (A, C)Loc-In 2 (A, C)Loc-In 12 (A, C)

Some inferences supported by our theory Is_a(C, A)Is_a(A, C)Is_a(C, B)Is_a(B, C) PP 1 (A, B)PP 1 (C, B)PP 1 (A, C) PP 2 (A, B)PP 2 (C, B)PP 2 (A, C) PP 12 (A, B)PP 1 (C, B)PP 2 (C, B)PP 2 (A, C)PP 1 (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 PP 1 1b Male Pelvis part_of Body PP 1 2 Cavity of Female Pelvis part_of Abdominal Cavity PP 1 3a Urinary Bladder part_of Female Pelvis PP 2 3b Urinary Bladder part_of Male Pelvis PP 2 4 Cell part_of Tissue PP 2 5 Right Ventricle part_of Heart PP 12 6 Urinary Bladder part_of Body PP 12 7 Nervous System part_of Body PP 12

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 PP 1 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 PP 2, NOT the inverse of PP 1 ). GALEN usually (but not always) asserts both A isDivisionOf B and B hasDivision A when PP 12 (A, B) holds. (note that PP 12 (A, B) is equivalent to PP 1 (A, B) & (PP -1 ) 1 (A, B).)

GALEN assertions using isDivisionOF and hasDivision GALEN’s isDivisionOf assertion BIT+Cl relation GALEN’s hasDivision BIT+Cl relation Female Pelvic Cavity isDivisionOf Pelvic Part of Trunk PP 1 none Prostate Gland isDivisionOf Genito- Urinary System PP 1 none Pelvic Part of Trunk hasDivision Hair (PP -1 ) 1 LeftHeartVentricle isDivisionOf Heart PP 12 Heart hasDivision LeftHeartVentricle (PP -1 ) 12 Prostate Gland isDivisionOf Male Genito-Urinary System PP 12 Male Genito-Urinary System hasDivision Prostate Gland (PP -1 ) 12 Urinary Bladder isDivisionOf Genito-Urinary System PP 12 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-In 1, Loc-In 2, or Loc-In 12. 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-In 1 2a Urinary Bladder contained_in Cavity of Female Pelvis Loc-In 2 2b Urinary Bladder contained_in Cavity of Male Pelvis Loc-In 2 3 Blood contained_in Cavity of Cardiac Chamber Loc-In 2 4 Urinary Bladder contained_in Pelvic Cavity Loc-In 12 5 Uterus contained_in Cavity of Female Pelvis Loc-In 12 6 Prostate contained_in Cavity of Male Pelvis Loc-In 12 7 Heart contained_in Middle Mediastinal Space Loc-In 12 8 Blood contained_in Lumen of Cardiovascular System Loc-In 12 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-In 1 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-In 2, NOT the inverse of Loc-In 1 ). GALEN usually (but not always) asserts both A isContaindIn B and B Contains A when Loc-In 12 (A, B) holds. (note that Loc-In 12 (A, B) is equivalent to Loc-In 1 (A, B) & (Loc-In -1 ) 1 (A, B).)

GALAN assertions using isContainedIn and Contains GALEN’s isContainedIn BIT+Cl relation GALEN’s Contains BIT+Cl relation 1 Ovarian Artery isContainedIn Pelvic Cavity Loc-In 1 Pelvic Cavity Contains Ovarian Artery (Loc-In -1 ) 2 2 Uterus isContainedIn Pelvic Cavity Loc-In 1 none 3 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-In 12 Female Pelvic Cavity Contains Uterus (Loc-In -1 ) 12 6 Mediastinum isContainedIn Thoracic Space Loc-In 12 Thoracic Space Contains Mediastinum (Loc-In -1 ) 12 7 Larynx isContainedIn Neck Loc-In 12 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 SubclassOf Is_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 R 1, R 2, and R 12 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).