Fuzzy mathematics in medicine
The book presents elements from fuzzy set theory needed to form a mathematical background for analysis, modeling and system design in medical practice and research. Chapter 1 briefly introduces set theory and relations, both crisp and fuzzy. Chapter 2 presents fuzzy relation equations which are a central model in computer-assisted medical diagnosis. Theoretical aspects, as well as some practical issues of acquisition of symptom-disease relationships are discussed. The reasoning mechanism implemented in the medical decision support system CADIAG-2 underlies this chapter. Fuzzy multi-criteria decision making is considered in Chapter 3. First, fuzzy outranking relations are introduced as the basis of preference modeling, detailing separately single-criterion and multiple-criteria cases. Next, group decision making and consensus are discussed. An example from genetics is used as an illustration. Bone mineral density has been studied by applying fuzzy preference modeling and decision making for analyzing the results found in 75 medical publications on this topic. Chapter 4 defines measures of fuzziness and fuzzy clustering, and describes medical applications thereof. Fuzzy c-means and a clustering model based on fuzzy equivalence relations are presented. The theory of fuzzy measures is introduced in Chapter 5. The authors proceed to briefly introduce evidence theory, belief functions and possibility theory. A view on medical diagnostic processes as an inference chain using degrees of belief or possibility measures is presented. The real-life example from Chapter 3 about bone mineral density is re-addressed. Possibility theory and fuzzy clustering are also exemplified on a stroke pathogenesis problem (used also in Chapter 4). Chapter 6 describes advanced concepts and ideas from fuzzy theory and their applications in medical diagnosis and treatment: fuzzy abductive inference for a diagnostic support system; fuzzy automata theory for an intensive-care clinical monitoring system; a fuzzified version of the so called ''genetic algebras''; and fuzzy hypergraphs for modeling activities of neuronal cell-assemblies. Four important algorithms from the text have been translated into C++ and attached as an Appendix: solving fuzzy relation equations; fuzzy clustering based on equivalence relations; calculating fuzzy measures; and combining two fuzzy measures.
scientific article; zbMATH DE number 1527758