INTRODUCTION

Risks of adverse outcomes are ubiquitous in the medical domain and have a central importance. An older population with complex combinations of chronic diseases and many medications makes simple deterministic treatment decisions difficult. Instead, clinicians need to assess, manage, and balance risks much more explicitly than ever before. It would therefore be valuable if ontologies aiming at adequately representing medical knowledge could formalize such risks. This paper contributes to this aim by proposing a framework for the representation of risks, illustrated by the risk of stroke in people with atrial fibrillation. More specifically, we propose a representation of absolute risks, such as a 3.2% risk of stroke over 12 months for people with atrial fibrillation (Nielsen et al., 2016).

This formalization is expressed in the Web Ontology Language (OWL), in the context of the OBO Foundry (Smith et al., 2007). The OBO Foundry is one of the most comprehensive collections of interoperable ontologies in the biomedical domain, built on the upper ontology Basic Formal Ontology (BFO) 2.0 (Arp, Smith & Spear, 2015). A few ontologies have formalized the notion of medical risk; see Uciteli et al. (2016) for a recent account (though not in the context of the OBO Foundry), as well as a review of former accounts. However, there is currently no comprehensive account of the notion of risk in the OBO Foundry ontologies.

One principle of BFO is the strict separation between universals and their instances. We write names of instances as well as relations between instances in bold, and names of universals and defined classes in italics. When first introduced, names of universals will be prefixed by the name of the source ontology (e.g., "BFO:Disposition"), unless the context makes it obvious.

The OBO Foundry compliant Ontology of Biological and Clinical Statistics (OBCS; Zheng et al., 2016) defines Absolute risk as a subclass of IAO:Information content entity ("ICE" for short). However, arguably, a person with atrial fibrillation has a risk to get a stroke independently of whether or not there exists some ICE estimating his risk to get a stroke. The risk itself has rather a dispositional character: an instance of risk of an adverse outcome of type A may be realized by an instance of A, but it may also never be realized; however, whether it is realized or not, the risk still exists. For this reason, this paper formalizes risks as dispositions that can be estimated by a specific kind of ICE, risk estimates, and the risk probability values as assigned to these risk estimates.

We begin by distinguishing two types of dispositions: the disease of atrial fibrillation on the one hand, and the risk of stroke of a human who has atrial fibrillation on the other (Sect. 2). We then show how a probability can be assigned to a risk of stroke in 12 months for a human with atrial fibrillation (Sect. 3). A discussion and conclusion follow.

DIFFERENTIATING RISK DISPOSITION AND DISEASE

The OGMS (Ontology for General Medical Science) considers a Disease as a BFO:Disposition (Scheuermann, Ceusters & Smith, 2009). Röhl & Jansen (2011) developed an axiomatisation of dispositions in the context of BFO. In this model, a disposition is a BFO:Dependent continuant that in-heres_in his bearer (which is the bearer_of this property), a Material entity, and may be realized (realized_in) via a process. The realization process has the material entity as a participant (has_participant), and the disposition is triggered by (has_trigger) some event or process. Finally, according to BFO, a disposition has_material_basis some entity. For example, the fragility of a glass is formalized as a disposition inhering in the glass, that may be realized by a breaking process when some form of stress (the trigger) happens; moreover, the fragility exists because of some molecular structure of the glass, which is its material basis.

The OGMS model considers Disease as a disposition re-alized_in a Disease course that has as parts some Pathological process. The material basis of a disease is a Disorder in the organism. For example, the disease epilepsy is seen as a disposition to have a disease course composed by various epileptic crises (pathological processes), because of some disorder in the brain.

The OGMS model is applied to cardiovascular diseases by the Cardiovascular Disease Ontology (CVDO; Barton et al., 2014). It formalizes the atrial fibrillation disease Atrial fibrillation as a disposition realized by a disease course that has as parts some processes of atrial fibrillation: This matches to an ambiguity in the natural language term "atrial fibrillation": it refers sometimes to a pathological process of atrial fibrillation (namely, irregular, uncoordinated contractions of the atria of the heart), and sometimes to a disease -a disposition exceeding a given threshold (Scheuermann et al., 2009) to atrial fibrillation processes.

Consider a human Jones, who has an atrial fibrillation disease ("AF" for short) af Jones -an instance of the universal Atrial fibrillation. Jones is an instance of Human AF , the class of humans with atrial fibrillation:

Human AF equivalentClass Human and (bearer_of some Atrial fibrillation) Suppose that af Jones leads to the process instance stroke-Jones (an instance of the universal Stroke) through the following scenario. There is some fibrosis in Jones' atrial myocardium (the disorder atrium_fibrosis Jones ), which is the material basis of the disposition af Jones : These pathological processes lead to the development of a blood clot in Jones' atrium, which is a new disorder. This blood clot is the bearer of a disposition to dislodge and migrate, which is at some point realized by the process of the blood clot migrating to the brain. The migrating clot has then a disposition to get stuck in a cerebral artery -by contrast to dissolve. When it gets stuck, the blood flow is blocked and stroke Jones happens.

af

On top of the disposition af Jones , Jones is also the bearer of another disposition: the risk of stroke risk Jones,Stroke , which is realized by his stroke: An instance of Risk AF,Stroke may be realized by one or several instance(s) of stroke, or may remain unrealized. To clarify the triggers of Risk AF,Stroke , we need to use BFO:History. BFO defines the history of a material entity as the "process that is the sum of the totality of processes taking place in the spatiotemporal region occupied by a material entity or site, including processes on the surface of the entity or within the cavities to which it serves as host" (Arp, Smith and Spear, 2015). We define History-part as the class of temporal parts of the history of any material object:

History-part equivalentClass (part_of some History)

In our formalization, Risk AF,Stroke is triggered by any History-part of its bearer:

Risk AF,Stroke subClassOf (has_trigger some History-part)

This way, risks of stroke are dispositions that are always triggered. However, Risk AF,Stroke is not a sure-fire disposition (that is, a disposition that is always realized when triggered), but a tendency (that is, a disposition that is not always realized when triggered ;Jansen 2007;Röhl & Jansen, 2011).

The material basis of risk Jones,Stroke is a disorder that has as part Jones' fibrosis, but also other entities. As a matter of fact, Jones can have a stroke because of his atrial fibrillation, but also because of various random or progressive factors, such as the regular senescence of his blood vessels.

risk Jones,Stroke has_material_basis some (Disorder and has_part atrium_fibrosis Jones and has_part senes-cent_blood_vessels Jones ) Thus, the disease af Jones and the risk risk Jones,Stroke are not the same disposition: even if they both inhere in Jones, they have distinct material basis and distinct realizations -namely, af_course Jones vs. stroke Jones . (The OGMS model leaves open the question whether pathological processes that are caused -or partially caused -by earlier pathological process of a disease course are also part of this disease course; thus, it is an open question whether stroke Jones is a part of af_course Jones ; see Barton et al. 2014 for a discussion. In any case, those two instances are distinct entities.)

PROBABILITY ASSIGNMENTS TO RISK DISPOSITIONS

We can now turn to the representation of a probability assignment to a risk of stroke. For this purpose, we first discuss the entity characterized by a probability assignment, then discuss the nature of probabilities at play, and finally formalize the probability assignment to a risk estimate.

What kind of entity do risk probabilities characterize?

Nielsen et al. (2016) studied a nationwide cohort of patients for which the overall ischemic stroke rate was 3.20 per 100 person-years. However, this value 3.2% does not relate only to a proportion in this cohort: we can infer from this information that an unspecified human from the same population with atrial fibrillation has a 3.2% probability to have a stroke over 12 months -even, of course, if he was not one of the patients in the cohort. Therefore, as we will now argue, the probability 3.2% also characterizes a certain property of Human AF : their risk to have a stroke over 12 months. Dispositions are natural targets for probability assignments (Barton et al. 2012). However, we cannot assign the probability 3.2% to the disposition Risk AF,Stroke . As a matter of fact, 3.2% is the probability of a human with AF to have a stroke over 12 months -but Risk AF,Stroke does not have any ontological connection with 12-months-long processes.

To solve this issue, let's define History-part 12m as the subclass of History-part with a 12 months-long duration. Let's now introduce Risk AF,12m,Stroke the class of risks of a human with AF to get a stroke over 12 months, that we also formalize as a Risk -and therefore, a disposition: In order to determine how a probability can characterize Risk AF,12m,Stroke , we need to clarify the ontological status of probabilities.

Risk AF,

What are the probabilities?

Standardly, objective and epistemic interpretations of probabilities are distinguished (Hájek, 2012). Objective probabilities are meant to characterize the world independently of our knowledge of it, while epistemic interpretations consider probabilities to describe our knowledge of the world: epistemic probabilities can be defined as degrees of belief or degrees of confidence.

Consider for example a biased coin that has three times more chances to fall on heads than on tails. The objective probability of the coin falling on heads is ¾, and its objective probability to fall on tails is ¼. If Mr. Green knows about the coin's bias, he should assign epistemic probabilities with the same values: ¾ to heads and ¼ to tails; this is a consequence of a principle of rationality called the "principal principle" (Lewis, 1980). However, if Mr. White is not aware of this bias and thinks that the coin is balanced, he would assign epistemic probabilities ½ to heads and ½ to tails. Epistemic probabilities can be operationalized as rational, hypothetical betting coefficients -that is, coefficients indicating which odds should be considered as acceptable by the agent to bet on the occurrence of heads or tails (Maher, 1997).

Suppose that the 3.2% value would be an objective probability that could be assigned to the disposition Risk AF,12m,Stroke . We would then have:

Risk AF,12m,Stroke subClassOf (has_objective_probability 0.032)

This would imply that for every r instance_of Risk AF,12m,Stroke : r has_objective_probability 0.032 Thus, all people with AF would have an objective probability 3.2% to have a stroke over 12 months. However, this cannot be the case: many people with AF have a lower or higher objective probability to have a stroke over 12 months, depending on various factors (see below the section 4.2 on CHADS 2 and CHADSVASC scores). Therefore, we would rather interpret 3.2% as an epistemic probability that characterizes the rational degree of confidence, given the evidence provided by Nielsen et al. (2016), that a person with AF would have a stroke over 12 months. We will now propose a formalization along those lines.

Epistemic probability assignment

We define the relation object_of as the inverse of IAO:is_about (Ceusters & Smith, 2010) To represent the assignment of the probability 3.2% to this risk estimate, we use two OBI relations (which are currently being formalized by the OBI development team), the object property has_value_specification (that relates an information content entity to an OBI:Value specification) and the datatype property has_specified_value (that relates a value specification to its numerical value). As a shortcut, let's introduce here the datatype property has_value defined as has_value_specification o has_specified_value. We can then write: risk_estimate AF,12m,Stroke,Nielsen_et_al._(2016) has_value 0.032 Informally, if R is a risk and re is a risk estimate, (R sub-ClassOf object_of re) and (re has_value p) mean together that according to the risk estimate re, it is rational to assign an epistemic probability p to the risk R.

The evidence for the estimate is documented in Niel-sen_et_al._(2016), an instance of IAO:Journal article (which is, in turn, a subclass of IAO:Document). To formally relate this journal article with risk_estimate AF,12m,Stroke,Niel- sen_et_al._(2016) , we introduce a relation extracted_from, whose domain is Information content entity and whose range is Document. If r is a risk estimate and j is a document, r ex-tracted_from j implies that j participates in a IAO:Planned process whose specified output is r: Altogether, the relations we have introduced here and in section 3.1 mean that a risk estimate is extracted from Nielsen et al. (2016), according to which it is rational to assign a 3.2% epistemic probability to the risk of stroke over 12 months for 1 As in OWL the term "value" is used in a class restriction to introduce an individual after an object property, this could more specifically be written as a person who has AF (in the absence of additional information).

extracted_from subRelationOf (is_specified_output_of o has_participant)

DISCUSSION

The reference class problem

As mentioned in section 3.2, all patients with atrial fibrillation do not have the same objective probability of stroke. The CHADS 2 score (congestive heart failure, hypertension, age ³ 75 years, diabetes mellitus, stroke) is a tool that has been developed to predict the risk of stroke in patients with atrial fibrillation by stratifying patients into risk groups (Gage et al., 2001). It was later expanded into the CHA 2 DS 2 -VASc score (Lip et al., 2010; written "CHADSVASC" from now on), which includes three additional risk factors: vascular disease, age 65-74 years, and female sex.

Let Human AF2 be the class of humans with atrial fibrillation and a CHADSVASC score of 2 ("AF2" for short). We can introduce a class of dispositions Risk AF2,12m,Stroke , the risk of stroke over 12 months for people with AF2: Risk AF2,12m,Stroke equivalentClass (Risk AF,12m,Stroke and inheres_in some Human AF2 ) Nielsen et al. (2016) state that the rate of stroke over 12 months among patients in the sample who had AF2 was 1.97%. Therefore, there is an instance risk_estimate AF2,12m,Stroke,Nielsen_et_al._(2016) such that: Risk AF2,12m,Stroke subClassOf object_of risk_estimate AF2,12m,Stroke,Nielsen_et_al._(2016) risk_estimate AF2,12m,Stroke,Nielsen_et_al._(2016) has_value 0.0197 Suppose that Jones has AF2. Jones is the bearer of the risk to get a stroke over 12 months risk Jones,12m,Stroke . Since Jones instance_of Human AF2 , his risk to get a stroke over 12 months is an instance of the class of risks to get a stroke over 12 months for people with AF2: risk Jones,12m,Stroke instance_of Risk AF2,12m,Stroke and therefore: risk Jones,12m,Stroke object_of risk_estimate AF2,12m,Stroke,Nielsen_et_al._(2016) Moreover, since Human AF2 subClassOf Human AF , Jones instance_of Human AF . Therefore, his risk to get a stroke over 12 months is an instance of the class of risks to get a stroke over 12 months for people with AF: risk Jones,12m,Stroke object_of risk_estimate AF,12m,Stroke,Nielsen_et_al._(2016) Thus, risk Jones,12m,Stroke is the object of two different estimates with two different probability values (0.032 and 0.0197), based on two different reference classes (AF or AF2): this is the reference class problem (Hájek, 2007). This is ontologically sound: if Dr. Khan only knows that Jones has AF, it is rational for him, based on Nielsen et al. (2016), to assign a probability 3.2% to the risk that Jones will have a stroke over 12 months; and if Dr. Patel knows in addition that Jones has a CHADSVASC score of 2, then it is rational for him, based on Nielsen et al. (2016), to assign a probability of 1.97% to this risk.

However, this raises practical difficulties. It might seem at first sight rational, for a computer system who has the information that Jones has a CHADSVASC score of 2, to always give precedence to the 1.97% risk estimation over the 3.2% estimation, as it is based on more specific factors. However, other criteria may matter. For example, if both values had been obtained from different studies, the 3.2% could be considered as a more reliable value for other reasons -such as a smaller 95% confidence interval.

Moreover, different articles relating about different cohorts or samples might give risk estimates with different values of the same risk class. They might give also risk estimates for risk classes that are not included into each others. Suppose that Jones has atrial fibrillation and is a smoker, and that we know two data from two different cohorts: the probability p AF that someone with atrial fibrillation will have a stroke during ten years; and the probability p Smoker that a smoker will have a stroke during ten years. There is no easy way to decide which epistemic probability is the best to estimate Jones' risk, or how they should be weighted in a common probability estimate. Note however that this is a classical issue for probabilistic reasoning, independent of the ontological representation chosen here.

Articulating objective and epistemic probabilities

We have seen earlier that we could not formalize in OWL 3.2% as an objective probability assigned to Risk AF,12m,Stroke , as it would imply that every instance of this risk (risk Jones,12m,Stroke , risk Hubbard,12m,Stroke , etc.) would have the same objective probability -which is false.

An alternative reading would be to interpret the objective probability 3.2% in line of Barton, Burgun & Duvauferrier (2012) as assigned only to the universal Risk AF,12m,Stroke , but not to its instances -a conception that is not straightforwardly implementable in OWL. Informally, this assignment would be elucidated as follows: in a hypothetical, representative sequence seq 0 of instances of Risk AF,12m,Stroke inhering in hypothetical instances of Human AF , the proportion of those risks who are realized -that is, the proportion of those humans who have a stroke over 12 months -tends towards 0.032 as the size of the sequence tends towards infinity.

This conception raises the issue of what it means to have a representative sequence of hypothetical instances of Risk AF,12m,Stroke inhering in hypothetical instances of Human AF . Indeed, several factors can influence the risk of having a stroke -in particular those involved in the CHADSVASC score: hypertension, age ³ 75 years, etc. It makes sense to speak of a representative sequence of instances only by reference to an actual population pop 0 : to be representative of pop 0 , the sequence seq 0 should involve the same proportions as in pop 0 of people with hypertension, of people older than 75 years, etc. But this implies that the probability 0.032 would characterize the actual population pop 0 (that is, a collection of human particulars -cf. Jansen & Schulz, 2011) rather than a subclass of Human. This is a possible orientation, pursed by Barton, Ethier, Duvauferrier & Burgun (2017) to formalize indicators of diagnostic performance.

This article has used an alternative interpretation of probabilities as epistemic in nature. This interpretation makes the formalization simpler in the present context, as it enables to relate a probability estimate to a universal of human. Future work will need to discuss further the articulation between the objective and epistemic probability views, and compare the strengths of each.

Generalization of this formalization

Note that this formalization can be adapted to represent a risk during a process that is not characterized by its duration (such as 12 months), but by some other characteristics. Imagine for example that we want to represent the probability p that a human with AF would have a stroke during a hospitalization process; we would then introduce the risk risk Jones,Hospitaliza- tion,Stroke that Jones would have a stroke during a hospitalization process, and assign the probability p to its risk estimate (the class of triggers of this risk would be the class of historyparts of Jones that temporally span any of his hospitalization process).

CONCLUSION

This article has shown how a specific risk and its various probability estimates could be formalized in the context of the OBO Foundry. We took the example of the risk of stroke for people with AF over 12 months Risk AF,12m,Stroke , which was formalized as a disposition. The article introduced risk_estimate AF,12m,Stroke,Nielsen_et_al._(2016) This representation of risk of stroke for patients with atrial fibrillation could also be used to stratify patients into risk groups by computing their CHADSVASC score, using e.g. SWRL rules (Rosier, 2015). This would also provide formal definitions of classes Human AF1 , Human AF2 , etc. This paper has shown how a specific example of probability assignment to a risk -the risk of stroke over 12 months of a patient with atrial fibrillation -could be formalized. Future work will need to systematize, using OBO-Foundry relations, the relations involving the classes Risk or Risk estimate. Elaborating on the work of Barton & Jansen (2016), a relation of disposition-parthood could also be introduced to represent the connection between the risk of a human with AF to have a stroke (Risk AF,Stroke ) and the risk of a human with AF to have a stroke over 12 months (Risk AF,12m,Stroke ).

The formalization presented in this paper relies on two hypotheses. The first is that for every class of material objects O, and every classes of processes T and R, there exists a class of dispositions D that inheres in O, has T as maximally specified class of triggers and R as maximally specified class of realizations; and O, T and R together constitute the conditions of identity of this disposition. This hypothesis was used to define Risk AF,12m,Stroke (from the classes Human AF , Historypart 12m and Stroke) as a class different from Risk AF,12m , which has a different maximally specified class of triggers (Historypart). The second hypothesis is that a realization r of a disposition d can happen during a trigger t -not necessarily just after the trigger ended. Thus, risk Jones,12m,Stroke could be realized during a 12-months-long history part that acted as a trigger. Finally, this formalization raises the wider philosophical issue whether the risk of stroke Risk Human,Stroke could be classified as a kind of disease, given OGMS definition of disease.