Using Cellular Automata Experiments to Model Periodontitis: a First Step towards Understanding the nonlinear Dynamics of the disease

Cellular automata (CA) are time and space discrete dynamical systems that can model biological systems. The aim of this study is to simulate by CA experiments how the disease of periodontitis propagates along the dental root surface. Using a Moore neighborhood on a grid copy of the pattern of periodontal ligament fibers (PDLF) supporting and anchoring the teeth to bone, we investigate the fractal structure of the associated using all possible outer-totalistic CA rules. On the basis of the produced propagation patterns, CA rules are classified in 3 groups, according to whether the disease was spreading, remaining constant or receding. These are subsequently introduced in a finite state Markov model as probabilistic “state-rules” and the model is validated using datasets retrieved from previous studies. Based on the maximum entropy production principle, we identified the “state-rule” that most appropriately describes the PDLF pattern, showing a power law distribution of periodontitis propagation rates with exponent 1.3. Entropy rates and mutual information of Markov chains were estimated by extensive data simulation. The scale factor of the PDLF used to estimate the conditional entropy of Markov chains was seen to be nearly equal 1.85. This possibly reflects the fact that a dataset representing tooth percentage with bone loss equal to 50% or more of their root length, is found to have a fractal dimension (FD) of about 1.84. Similarly, datasets of serum neutrophil, basophil, eosinophil, monocyte counts and IgG, IgA, IgM levels taken from periodontitis patients, showed a FD ranging from 1.82 to 1.87. Our study presents the first mathematical model to our knowledge that suggests periodontitis is a nonlinear dynamical process. Moreover, the model we propose implies that the entropy rate of the immune-inflammatory host response dictates the rate of periodontitis progression. This is validated by clinical data and suggests that our model can serve as a basis for detecting periodontitis susceptible individuals and shaping prognosis for treated periodontitis patients.