Heart Failure Evolution Model Based on Anomalous Diffusion Theory

The unexpectable variations of the diagnosed disease symptoms are quite often observed during medical diagnosis. In stochastics, such behavior is called "grey swan" or "black swan" as synonyms of sudden, unpredictable change. Evolution of the disease's symptoms is usually described by means of Markov processes, where dependency on process history is neglected. The common expectation is that such processes are Gaussian. It is demonstrated here that medical observation can be described as a Markov process and is non-Gaussian. Presented non-Gaussian processes have "fat tail" probability density distribution (pdf). "Fat tail" permits a slight change of probability density distribution and triggers an unexpectable big variation of the diagnosed parameter. Such "fat tail" solution is delivered by the anomalous diffusion model applied here to describe disease evolution and to explain the possible presence of "swans" mentioned above. The proposed model has been obtained as solution of the Fractal Fokker-Planck equation (FFPE). The paper shows a comparison of the results of the theoretical model of anomalous diffusion with experimental results of clinical studies using bioimpedance measurements in cardiology. This allows us to consider the practical usefulness of the proposed solutions.