# Heart Failure Evolution Model Based on Anomalous Diffusion Theory

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Data

^{2}tests) but not too high to avoid complexity of calculations. With computer calculation, we estimate that the amount of 15 slots allows for an acceptable accuracy of calculations. The time series was registered as follows: Due to the irregular flow of the measured data per patient, we adopted a registration period that was 31 days long for each slot. Each slot includes hits counted during measurements registered over a five-month period. Longer registration does not carry new results in TFC distribution. Hits registered during each single measurement period are illustrated in Figure 1.

#### 2.2. Theoretical Models

_{α}to ensure proper pdf interpretation. The constraint applies just to solution classes in Equation (9). Now, one may simulate the obtained solution (9–11), which must obey the predefined constraint 0 ≤ P

_{α,Kα}(x,t) ≤ 1 for a set of pairs {α, K

_{α}}. The general constraint of 0 < α < 2 applies, as a rule, for fractional FFPE. The condition that P(x, t) is a decreased function of both arguments also imposes a constraint on the permissible pairs {α, K

_{α}}. Figure 2 presents an example of the simulated P

_{α,Kα}(x,t) with arbitrary values of α and K

_{α}.

## 3. Results

_{α}placed inside Figure 3 have been calculated by means of root mean square error (RMSE) between measured experimental data and the theoretical “fat tail” distribution presented in Equation (12), closed for a time of observation as long as 5 months. One can see that the Hurst exponent is equal to α/2 = H = 0.235.

## 4. Discussion

## Funding

## Conflicts of Interest

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**Figure 1.**Measured number of hits of TFC values for each period of registration. Items from t1 to t5 mean one month of hits registration indicated by bar color in the figure.

**Figure 3.**Comparison of TFC clinical observations (blue) and theoretical results (grey) for parameters α and K

_{α}minimizing RMSE error value (RMSE = 0.003).

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Walczak, A.A.
Heart Failure Evolution Model Based on Anomalous Diffusion Theory. *Entropy* **2022**, *24*, 1780.
https://doi.org/10.3390/e24121780

**AMA Style**

Walczak AA.
Heart Failure Evolution Model Based on Anomalous Diffusion Theory. *Entropy*. 2022; 24(12):1780.
https://doi.org/10.3390/e24121780

**Chicago/Turabian Style**

Walczak, Andrzej Augustyn.
2022. "Heart Failure Evolution Model Based on Anomalous Diffusion Theory" *Entropy* 24, no. 12: 1780.
https://doi.org/10.3390/e24121780