# Evolution of Hemodynamic Parameters Simulated by Means of Diffusion Models
## 4 Discussion and Conclusions  
Claiming that hemodynamic parameter values measured in a population of patients behave in accordance with the rules of drifted diffusion, we have identified a step-by-step procedure allowing to verify such a statement. From the green function of FPE, we obtained stationary distribution \(W(x)\) of the thoracic fluid content (TFC) parameter observed, and by minimizing RMSE between the theoretical model and the experimental observations of stationary distribution, we adjusted the parameters of the stochastic process with the use of conjugated gradient algorithms. It turned out that the resulting form of the stochastic process is the Ornstein-Uhlenbeck process. So, we found the Ornstein-Uhlenbeck diffusion process as a model for medical treatment of heart failures. From the analytical form of the FPE solution (green function of FPE), we determined the \"ex definition\" distribution \(P(x,t)\) for the observed hemodynamic parameters and determined dynamics of TFC evolution. The last step had the form of a statistical \(\chi^{2}\) test, aiming to accept or to reject the proposed theoretical model. The \(\chi^{2}\) test confirms the accepted level of efficiency of the theoretical model of the Ornstein-Uhlenbeck process, describing TFC evolution. Finally, we may conclude that the diffusion processes have turned out to be useful tools for predicting disease evolution. We have also proved that:  
1. The Ornstein-Uhlenbeck process, including the linear drift component, seems to be a precise tool for simulating the evolution of TFC and, supposedly, other hemodynamic parameters, also during medical treatment. 2. We obtained the distribution \(P(x,t)\), allowing us to predict and simulate TFC evolution. 3. Parameter \(\gamma\) describes the effectiveness of medical treatment in the population of patients, thus being a mathematical measure of such treatment effectiveness that is worth being analyzed in a more thorough manner, e.g., by focusing on variations depending on sex, occupation, age, severity of heart failure in accordance with the NYHA standard, etc. The idea of diffusion occurring inside a potential force field is deeply exploited in solid state physics and quantum electronics [16; 17; 18; 19; 20; 21; 22], but its successful application in medical diagnostics seems to be a new approach that suggests, to some extent, the existence of a \"hidden general symmetry\" between physics and biology as well. Diffusion theory is relevant not only for microscopic observations made in physics, but also for much larger scale phenomena, e.g., in medicine. In medicine, fluctuations take place due to multiple and multilevel interactions occurring in the patient's system. Such fluctuations are hard to describe in a precise manner, although their presence seems to be obvious. The results presented herein are based on data processed during a medical experiment and, to the best of the authors' knowledge, form the first model of its type proposed in medical diagnostics. The results we obtained allows us to predict the time evolution of TFC. Simultaneously, we have confirmed that the fluctuation theory, reflected here by means of FPE, is an inherent element of medical diagnostic processes. It must be underlined that prospective validation of the model is continued now in collaborative clinics under Amulet project. We do suppose that limitations of the model will be analyzed during further validation. An evident strength of the model is its continuous, differential model of disease evolution. As far as authors know, it seems to be a new approach.