Granularity Explains Empirical Factor-of-Three Relation Between Probabilities of Pulmonary Embolism in Different Patient Categories

Pulmonary embolism is a very dangerous difficult-to-detect medical condition. To diagnose pulmonary embolism, medical practitioners combine indirect signs of this condition into a single score, and then classify patients into low-probability, intermediate-probability, and high-probability categories. Empirical analysis shows that, when we move from each category to the next one, the probability of pulmonary embolism increases by a factor of three. In this paper, we provide a theoretical explanation for this empirical relation between probabilities. 1 Formulation of the Problem Pulmonary embolism: a brief reminder. One of the most dangerous medical conditions is pulmonary embolism, a blockage of the main artery of the lung (or one of its branches) which can lead to collapse and sudden death; see, e.g. [1]. Pulmonary embolism is responsible for about 15% of sudden deaths. If detected on time, pulmonary embolism can be treated: either by anticoagulation medicine like heparin or warfarin, or – in severe cases – by a surgery. The problem is that pulmonary embolism is difficult to diagnose: lungs are mostly normal, fever is either absent or low-grade, etc. Scores: a brief description. Since pulmonary embolism is difficult to directly diagnose, hospitals’ emergency departments take into account different variables