A Path-Probability Approach to Irreversible Markov Chains with an Application in Studying the Dental Caries Process

Abstract

SUMMARY By letting 1 and 0 denote the presence and absence of caries on a surface, the 32 possible states of caries pattern on a tooth may be represented by a five-bit binary number X1X2X3X4X5 where Xi = O 1, and Sk = k - 1. The caries process may be considered as an irreversible Markov chain with S, as the initial state and S32 as the absorbing state. The analysis of the irreversible chain may be approached by an algorithm based on the conditional matrix P instead of the fundamental matrix N. From matrix P, a useful path diagram may be constructed to obtain many Markov statistics and the probabilities associated with each observed pathway through which the caries process progresses. Three year data observed at six-month intervals on the maxillary second bicuspid were used as an example. It was found that the caries process occupied nine states: (00000), (10000), (01000), (00100), (10100), (11000) (01100), (11100), and (11111). The susceptibility of caries attack in descending order is distal, mesial, and occlusal. No lingual or buccal decay was observed. All tooth loss resulted from state (11100), the occlusal-mesial-distal decay. The three most important pathways are as follows: