Avicenna Model for Medical Diagnosis

The model of medical diagnosis in Avicenna tries to simulate diagnostic decision making as done by the physicians. Therefore, it is necessary to know how a physician, consciously or unconsciously, evaluates distinct diagnoses conditional on the basis of a given set of patient’s symptoms. According to the proposals from many experts, the physicians normally have an imperfect knowledge of how they solve diagnostic problems; but they usually follow some general principles when they are confronted with an actual case. In fact, when a physician is confronted with an actual case, he takes into consideration the strength of confirmation and the strength of disconfirmation (exclusion) of the patient’s symptoms with respect to distinct diagnoses. Strength of confirmation is the degree of the acceptance of a diagnosis conditional on the basis of a set of symptoms exhibited by the patient in comparison with other diagnoses. Correspondingly, strength of exclusion is the degree of the disconfirmation of a diagnosis on the basis of the (expected) symptoms that they are not exhibited by the patient. Every physician has his individual criteria to assess the strength of confirmation (and also, disconfirmation) of the patient’s symptoms, but the following criteria for the strength of confirmation are nearly accepted by all of the physicians:

  1. Knowledge Extension Principle: The more (less) diagnoses a physician associates with a certain symptom, the less (more) is the strength of confirmation of the symptom for a certain diagnosis. For example, the strength of confirmation of the ‘hepatomegaly’ is more than the ‘fever’; or equivalently, ‘hepatomegaly’ is a more specific symptom than ‘fever’.
  2. Frequency Principle I: The more (less) often a certain symptom occurs in a certain diagnosis, the more (less) is the strength of confirmation of the symptom for the diagnosis. For example, the strength of confirmation of the ‘sneezing’ is more than the ‘sore throat’ for the ‘common cold’; or equivalently, ‘sneezing’ confirms ‘common cold’ more than ‘sore throat’.
  3. Sufficiency Principle: If a symptom is observed in only one diagnosis, the existence of that symptom absolutely confirms the respective diagnosis. For example, the ‘trouble with counting and adding’ absolutely confirms the ‘mathematics disorder’.
  4. Monotonicity Principle I: If two symptoms confirm the existence of a diagnosis independently, the existence of both of them together, confirms the respective diagnosis more than the case that one of them confirms it. For example, ‘runny nose’ and ‘nasal congestion’ confirms the ‘common cold’ more than every one.

Also, the following criteria for the strength of disconfirmation are nearly accepted by all of the physicians:

  1. Domain Extension Principle: The more (less) symptoms occur with a certain diagnosis, the less (more) is the strength of exclusion of a not exhibited symptom for the respective diagnosis. For example, non existence of ‘vision problems’ makes ‘astigmatism’ more unbelievable than ‘diabetes’.
  2. Frequency Principle II: The more (less) often a certain symptom occurs in a certain diagnosis, the more (less) is the strength of exclusion for non existence of the respective symptom. For example, non existence of ‘sore throat’ makes ‘pharyngitis’ more unbelievable than (non existence of) ‘breathing difficulty’.
  3. Necessity Principle: If a symptom is always occurring with a diagnosis, and the symptom is not found in the patient, the respective diagnosis should be excluded. For example, if the ‘headache’ is not exhibited by the patient, ‘sinusitis’ should be excluded.
  4. Monotonicity Principle II: If non existence of two symptoms disconfirm a diagnosis independently, non existence of both of them disconfirm the respective diagnosis more than every one. For example, non existence of ‘runny nose’ and ‘nasal congestion’ disconfirms the ‘common cold’ more than the case that just one of them is missed.

The model of medical reasoning in Avicenna is based on the above proposed principles. In this model, a diagnosis is evaluated on the basis of a given set of symptoms, the overlap of the set of symptoms that caused by that diagnosis, in conjunction with the set of symptom which were given, and the influence of the not exhibited symptoms by the patient. Avicenna reasoning model can be stated as the following mathematical formulation:

Let sigma and delta be non-fuzzy sets of all symptoms and diagnoses, respectively. Let also R be a fuzzy relation denoting the frequency of occurrence of a certain symptom with a certain diagnosis. It is clear from the etiology of some (but not all) diseases that we can encode the clinicians’ knowledge of diagnosis-symptom relationships into frequency of occurrence relations with a high degree of accuracy. For example, the statement of “Chest infection always causes coughing” can be encoded into a fuzzy relation R such as:

Or, the statement of “A runny nose is usually present in a common cold” can be encoded into another fuzzy relation R', respectively:

Now let K be a set of all fuzzy relations which relate the diagnoses into the symptoms. K can then be considered as our medical knowledge base. Associate with each symptom , a fuzzy set of diagnoses:

that cause the symptom sigma, and associate with each diagnosis , a fuzzy set of symptoms:

that it causes, called the profile for that diagnosis. Let also
e(S)
and
c(D)
for sets S and D. Therefore, we define a diagnosis delta explains a set of symptoms S to some degree that it can satisfy the above mentioned principles. In accordance with our definition, we introduce several symbols as follows:

  1. R: Ranking Measure, or the degree of confirmation of a diagnosis delta on the basis of a given set of symptoms S compared to other diagnoses;
  2. E: Excluding Measure, or the degree of disconfirmation (exclusion) of a diagnosis delta on the basis of a given set of symptoms S that they were expected to be exhibited by the patient in disease delta, but they are not exhibited.
  3. C: Covering Measure, or the degree of covering of a given set of symptoms S by a diagnosis delta;

Ranking measure is motivated by the problem of how much a diagnosis is acceptable and covering measure (and similarly, excluding measure) is motivated by the problem of how to interpret the expected symptoms that they are not exhibited by the patient. Hence, the task of diagnosis in Avicenna will be reduced to calculating these two measures and then, collecting them into an acceptability measure for each diagnosis based upon a given (non-empty) set of symptoms. There are too many ways of defining and interpreting the ranking, excluding, and covering measures. We then define them in the unit interval (respectively) as follows:



and

Here, nu is a fuzzy t-conorm where  (absorbent element). Obviously, all of ranking, excluding, and covering measures belong to the unit interval. Both ranking and covering measures are fuzzy numbers; in the case that the value of 0 means that there is no proof (reason) for that respective measure, while the value of 1 is interpreted as a proof (definitive reason). These two values are independent from each other; however they act in a same direction. When both, ranking and covering measures, are equally 1, it is interpreted as a conclusive proof for the respective diagnosis, and correspondingly, when both of them have 0 values, it means that the respective diagnosis should be excluded. Intermediate values tell us that there is not sufficient reason to prove or exclude the diagnosis (we never encounter with a situation where covering = 1 and ranking = 0; but the situation where ranking = 1 and covering = 0 can be interpreted as a new case which is out of our knowledge base.) and the interpretation of a situation where both ranking and covering are not 1 and 0 will leave to the user. According to the above sentences, we make the Possibility Function  which describes the degree of acceptance (explanation) of a diagnosis delta conditional on a given set of symptoms S as following:

There are too many ways to define the possibility function and t-conorm operator, while all of them must be monotonic continuous functions in the unit interval. Any definition of possibility function and its t-conorm allow the user to evaluate the diagnoses according to the needs and decisions about the interpretation of the knowledge base; however, any choice of them will be compatible with our principles and hence, it can demonstrate how the physicians make diagnostic decisions.  Then we propose the following definitions for them:


where x and y are ranking and covering measures, respectively. It is completely obvious that the proposed possibility function and the corresponding t-conorm satisfy our conditions. In order to evaluate Avicenna reasoning ability, it was challenged to diagnose a series of actual patients each of whom had been referred to an internist and in which of whom a diagnosis had been established. All of the entered cases were real clinical cases and included all of the “noise” of the actual diagnostic evaluation. We just omitted the cases which their established diagnoses (according to the referring internist) were not present in the Avicenna knowledge base. The cases categorized according to the organ system or systems involved and translated into the language provided by the program. Because of the limitations of program’s language, some data could only be approximated, or could not be entered at all. After this validation stage, 250 selected cases were entered into software and the system produced a ranked list of possible diagnoses for each case. The results showed that the delivered diagnoses by the internist had obtained the maximum possibility (highest ranking) in 215 of 250 cases (sensitivity = 0.86), although approximately six most possible diagnoses per case appeared on the generated list by the program (low specificity). Also, it was showed that the mean length of symptoms was about five.