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PPT – A Formal Theory for Spatial Representation and Reasoning in Biomedical Ontologies PowerPoint presentation | free to download - id: 1bd7ff-ZDc1Z

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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
- 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-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 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

- 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 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

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 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

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

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

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.

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

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

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

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

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)

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).