Evolution of Hemodynamic Parameters Simulated by Means of Diffusion Models

Abstract

Diffusion is a well-known physical phenomenon governing such processes as movement of particles or transportation of heat. In this paper, we prove that a close analogy to those processes exists in medical data behavior, and that changes in the values of medical parameters measured while treating patients may be described using diffusion models as well. The medical condition of a patient is usually described by a set of discrete values. The evolution of that condition and, consequently, of the disease has the form of a transition of that set of discrete values, which correspond to specific parameters. This is a typical medical diagnosis scheme. However, disease evolution is a phenomenon that is characterized by continuously varying, temporal characteristics. A mathematical disease evolution model is, in fact, a continuous diffusion process from one discrete slot of the diagnosed parameter value to another inside the mentioned set. The ability to predict such diffusion-related properties offer precious support in diagnostic decision-making. We have examined several hundred patients while conducting a medical research project. All patients were under treatment to stabilize their hemodynamic parameters. A diffusion model relied upon simulating the results of treatment is proposed here. Time evolution of thoraric fluid content (TFC) has been used as the illustrative example. The objective is to prove that diffusion models are a proper and convenient solution for predicting disease evolution processes. We applied the Fokker-Planck equation (FPE), considering it to be most adequate for examining the treatment results by means of diffusion. We confirmed that the phenomenon of diffusion explains the evolution of the heart disease parameters observed. The evolution of TFC has been chosen as an example of a hemodynamic parameter.

Heart Failure Evolution Model Based on Anomalous Diffusion Theory

“…The results described in papers [ 6 , 7 ] indicated good agreement between experimental observations and theoretical models, confirming to some extent that the observed processes can be considered Gaussian processes. In the following, we examine anomalous diffusion to describe the distribution of observed hemodynamic parameters to show that the process may nevertheless have “fat tails.”…”

“…As mentioned, two theoretical models—the Markov process [ 6 ] and the analytical solution of the Fokker–Planck equation (FPE) of diffusion containing a drift component for modeling the presence of therapy [ 7 ]—were previously studied in parallel.…”

“…We have proposed in earlier publications a different analysis using heart failure cases as an example [ 6 , 7 ]. Since we do not have a very large population of measurements of the selected parameter to be diagnosed (a few thousand measurements made under the same conditions of clinical observation of the patient, which guarantees their statistical utility but is a relatively small population), we checked whether, in a smaller population, the time series of observations is or is not well approximated by a Markov process.…”