Genesis, Growth, and Therapy of Tumors: Mathematical Models

Many phenomena in biology and medicine display regularities which can be described mathematically (though quite often in a particular set of observations these regularities are rather well hidden behind irregularities introduced by factors unrelated to the phenomenon observed, i.e. by »biological variation« and »random fluctuations«). Mathematical models are constructed not only to describe and explain quantitative relationships but also to predict aspects of the phenomena which are not yet observed or even unobservable.

Tumor research is one of the areas in which mathematical work of this kind has been done in about the last two decades. Recent developments have been stimulated by the special sessions of the fourth and fifth Berkeley Symposia devoted to the subject. The purpose of the present paper is to give a survey over the work that has been done and to indicate current trends. An attempt has been made to provide the reader with a complete list of references.

Viele biologische und medizinische Erscheinungen zeigen Regelmäßigkeiten, die einer mathematischen Beschreibung zugänglich sind (allerdings werden diese Regelmäßigkeiten häufig überlagert von »biologischer Variation« und »zufälligen Schwankungen«, die in keiner direkten Beziehung zu der beobachteten Erscheinung stehen). Mit Hilfe mathematischer Modelle will man nicht nur quantitative Beziehungen beschreiben und verstehen, sondern auch Aussagen gewinnen über bisher nicht beobachtete oder gar der Beobachtung nicht zugängliche Aspekte des betrachteten Geschehens. Ein Gebiet, in dem seit etwa zwei Jahrzehnten in dieser Richtung’mathematisch gearbeitet wird, ist die Krebsforschung. Auf den letzten beiden Berkeley-Symposien über »Mathematische Statistik und Wahrscheinlichkeitstheorie« wurde diesem Anwendungsgebiet viel Aufmerksamkeit geschenkt. Diese Arbeit soll einen Überblick über den Stand der Entwicklung mathematischer Modelle in der Tumorforschung geben. Im Schriftenverzeichnis wurde versucht, die vorliegende Literatur möglichst vollständig zu erfassen.

• References

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