Mathematical Background of Medical Diagnosis

There are several ways to formalize the medical diagnosis process using mathematical models; but one of the most general theories used to describe foundation of medical diagnosis is the set-covering theory. This theory can be stated as the following mathematical formulation:

Let S be a set of symptoms and let D be a set of diseases. Let also K be a binary relation associating each disease in D with symptoms in S that are caused by that disease. K can then be considered to be a knowledge base. Associate with each symptom s, a set of diseases causedby that cause the symptom, and associate with each disease d, a set of symptoms causes that it causes, called the profile for that disease. Let also CousedbyCouses for sets  and . Therefore, the task of diagnosis can be stated as the search for a set of diseases that, ‘explain’ a set of symptoms exhibited by the patient. By viewing this problem in the context of set theory, a disease d explains a symptom s if it causes it, i.e. s. A differential diagnosis can thus be obtained for a set S' of symptoms, by selecting all diseases that explain a symptom in S', i.e. the set CausedBy. For example, given the knowledge base:

K

the set of all diagnoses D' for the set S' are:

D'

 

This set is the least refined explanation that we can produce using the knowledge base, and in general is not a particularly helpful one. A set of disease D is said to be an explanation of a diagnostic problem with observed symptoms S if D and it satisfies some additional criteria. Various criteria, in particular so-called criteria of parsimony, are in use. The basic idea is that among the various diagnoses of a set of symptoms, those that satisfy certain criteria of parsimony are more likely than others. Some of these criteria are:

  1. Minimal cardinality: a diagnosis D is an explanation if and only if it contains the minimum number of elements among all diagnoses;
  2. Irredundancy: a diagnosis D is an explanation if and only if no proper subset of D is a diagnosis;
  3. Relevance: a diagnosis D is an explanation if and only if D;
  4. Most probable diagnosis: a diagnosis D is an explanation if and only if 

According to above criteria, the explanation of our example will reduce to:

D