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).