# Fractional-Order Windkessel Boundary Conditions in a One-Dimensional Blood Flow Model for Fractional Flow Reserve (FFR) Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Coronary Blood Flow Model

#### 2.2. Fractional-Order Boundary Conditions

#### 2.3. Model Personalization

#### 2.4. Patient Data

## 3. Results

#### 3.1. Blood Pressure and FFR Sensitivity to Order $\alpha $

#### 3.2. Patient-Specific Calculations

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CCTA | coronary computed tomography angiography |

DA | diagonal artery |

FFR | fractional flow reserve |

HR | heart rate |

LCA | left coronary artery |

LCx | left circumflex artery |

LAD | left anterior descending artery |

LADd | distal part of the left anterior descending artery |

LADp | proximal part of the left anterior descending artery |

RCA | right coronary artery |

RMSE | root mean square error |

SV | stroke volume |

## Appendix A

**Table A1.**Parameters of the vessels for simplified structure (Figure 1).

Segment № | Length, cm | Diameter, mm | c, m/s |
---|---|---|---|

1 | 3.2 | 22.0 | 7.5 |

2 | 5.0 | 25.0 | 7.5 |

3 | 1.3 | 3.9 | 9.0 |

4 | 6.0 | 2.8 | 9.0 |

5 | 3.0 | 3.0 | 9.0 |

6 | 0.7 | 0.9 | 9.0 |

7 | 4.0 | 0.3 | 9.0 |

8 | 7.5 | 3.0 | 9.0 |

**Table A2.**Parameters of the vessels for Patient 1 (Figure 4).

Segment № | Length, cm | Diameter, mm | c, m/s |
---|---|---|---|

1 | 3.1 | 21.0 | 9.7 |

2 | 5.0 | 23.0 | 9.7 |

3 | 1.3 | 3.9 | 11.6 |

4 | 2.3 | 2.3 | 11.6 |

5 | 1.1 | 0.7 | 11.6 |

6 | 1.0 | 2.6 | 11.6 |

7 | 2.7 | 1.4 | 11.6 |

8 | 3.9 | 1.9 | 11.6 |

9 | 2.1 | 3.1 | 11.6 |

10 | 1.8 | 2.0 | 11.6 |

11 | 6.8 | 2.0 | 11.6 |

12 | 7.6 | 1.4 | 11.6 |

13 | 4.6 | 1.7 | 11.6 |

14 | 1.1 | 1.3 | 11.6 |

15 | 7.5 | 1.3 | 11.6 |

**Table A3.**Parameters of the vessels for Patient 4 (Figure 4).

Segment № | Length, cm | Diameter, mm | c, m/s |
---|---|---|---|

1 | 2.9 | 21.0 | 8.8 |

2 | 5.0 | 22.0 | 8.8 |

3 | 0.6 | 2.4 | 10.6 |

4 | 0.9 | 3.0 | 10.6 |

5 | 2.3 | 1.3 | 10.6 |

6 | 2.2 | 2.0 | 10.6 |

7 | 9.7 | 1.9 | 10.6 |

8 | 2.8 | 1.3 | 10.6 |

9 | 4.7 | 1.8 | 10.6 |

10 | 3.0 | 3.0 | 10.6 |

11 | 6.5 | 2.2 | 10.6 |

12 | 9.4 | 2.8 | 10.6 |

13 | 1.2 | 1.6 | 10.6 |

14 | 4.6 | 1.3 | 10.6 |

15 | 9.9 | 1.7 | 10.6 |

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**Figure 1.**A simplified network of major coronary arteries. Segment 6 represents 66% stenosis. The model parameters for each segment are presented in Table A1 in Appendix A. We impose cardiac output function (Figure 2) on the inlet of segment 1. On the terminal ends of segments 4, 7, and 8, we impose hydraulic resistance and outflow pressure (6). Boundary condition on the terminal end of the aorta (segment 2) involves a 2-element Windkessel model (7).

**Figure 5.**Aortic blood pressure for various orders $\alpha $. (

**a**) All parameters, except for $\alpha $, are fixed. (

**b**) Model parameters were adjusted to yield the same values of the systolic and diastolic blood pressure.

**Figure 6.**Mean pressure and FFR for a simplified network of coronary arteries. (

**a**) Mean pressure and $\alpha $. (

**b**) FFR and $\alpha $. The horizontal red line corresponds to FFR = 0.8—threshold value between significant (FFR < 0.8) and insignificant (FFR > 0.8) stenoses.

**Table 1.**Characteristics of the patient dataset (mean ± standard deviation). Details are presented in [5]. $\theta =\frac{{P}_{mean}-{P}_{dia}}{{P}_{sys}-{P}_{dia}}$ is a measure of blood profile thickness.

Characteristic | Value |
---|---|

Number of patients | 9 |

Number of males | 5 |

Heart rate, bpm | 69 ± 14 |

Systolic pressure ${P}_{sys}$, mm Hg | 141 ± 23 |

Diastolic pressure ${P}_{dia}$, mm Hg | 73 ± 8 |

Mean pressure ${P}_{mean}$, mm Hg | 103 ± 11 |

BMI, kg/m${}^{2}$ | 29 ± 4 |

$\theta $ | 0.45 ± 0.10 |

**Table 2.**Characteristics of Patient 1 and Patient 4. Stroke volume (SV) was estimated from patient age and BMI values presented in [5].

Patient 1 | Patient 4 | |
---|---|---|

Age, years | 80 | 68 |

HR, bpm | 67 | 88 |

SV, mL | 82 | 70 |

${P}_{sys}$, mm Hg | 174 | 130 |

${P}_{dia}$, mm Hg | 76 | 66 |

${P}_{mean}$, mm Hg | 111 | 94 |

Stenosis location | LAD | LAD |

Stenosis degree | 70% | 60% |

FFR measured | 0.89 | 0.82 |

**Table 3.**FFR estimations for the patient dataset. Patient data: ${P}_{mean}$ is the measured mean pressure, mm Hg; $\theta $ is a measure of the pressure profile thickness (14); Loc. is a location of stenosis; $FFR$ is the invasively measured FFR. The original approach for FFR estimation (order $\alpha =1.0$): $FF{R}_{\alpha =1}$ is the calculated FFR value with a boundary condition (7); ${P}_{mean}^{est}$ is the calculated mean pressure with order $\alpha =1$. The fractional derivative approach (order $\alpha ={\alpha}_{opt}$) involves adjusting order $\alpha $ so that the calculated mean pressure matches the measured one: $FF{R}_{\alpha ={\alpha}_{opt}}$ is the calculated FFR value with the boundary condition (9), and fractional order $\alpha ={\alpha}_{opt}$; ${\alpha}_{opt}$ is the optimal fractional order. Patient 9 was excluded due to the absence of a mean pressure measurement.

Patient Data | Order $\mathit{\alpha}=1.0$ | Order $\mathit{\alpha}={\mathit{\alpha}}_{\mathit{opt}}$ | ||||||
---|---|---|---|---|---|---|---|---|

№ | ${\mathit{P}}_{\mathit{mean}}$ | $\mathit{\theta}$ | Loc. | $\mathit{FFR}$ | ${\mathit{FFR}}_{\mathit{\alpha}=1}$ | ${\mathit{P}}_{\mathit{mean}}^{\mathit{est}}$ | ${\mathrm{FFR}}_{{\alpha}_{\mathrm{opt}}}$ | ${\alpha}_{\mathrm{opt}}$ |

1 | 111 | 0.36 | LAD | 0.89 | 0.83 | 118 | 0.87 | 0.56 |

2 | 83 | 0.46 | LAD | 0.86 | 0.87 | 83 | 0.87 | 1.0 |

3 | 125 | 0.40 | RCA | 0.88 | 0.89 | 125 | 0.89 | 1.0 |

4 | 94 | 0.44 | LAD | 0.82 | 0.82 | 94 | 0.82 | 1.0 |

5 | 99 | 0.39 | LAD | 0.82 | 0.8 | 102 | 0.82 | 0.82 |

6 | 99 | 0.40 | LADp | 0.9 | 0.98 | 101 | 0.98 | 0.91 |

LADd | 0.82 | 0.87 | 0.88 | |||||

DA | 0.81 | 0.84 | 0.85 | |||||

7 | 98 | 0.37 | LAD | 0.75 | 0.68 | 102 | 0.71 | 0.78 |

LCx | 0.84 | 0.82 | 0.85 | |||||

8 | 110 | 0.51 | LAD | 0.88 | 0.92 | 107 | 0.92 | 1.3 |

LCx | 0.89 | 0.97 | 0.96 | |||||

10 | 108 | 0.51 | LAD | 0.72 | 0.81 | 90 | 0.8 | 1.85 |

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**MDPI and ACS Style**

Gamilov, T.; Yanbarisov, R.
Fractional-Order Windkessel Boundary Conditions in a One-Dimensional Blood Flow Model for Fractional Flow Reserve (FFR) Estimation. *Fractal Fract.* **2023**, *7*, 373.
https://doi.org/10.3390/fractalfract7050373

**AMA Style**

Gamilov T, Yanbarisov R.
Fractional-Order Windkessel Boundary Conditions in a One-Dimensional Blood Flow Model for Fractional Flow Reserve (FFR) Estimation. *Fractal and Fractional*. 2023; 7(5):373.
https://doi.org/10.3390/fractalfract7050373

**Chicago/Turabian Style**

Gamilov, Timur, and Ruslan Yanbarisov.
2023. "Fractional-Order Windkessel Boundary Conditions in a One-Dimensional Blood Flow Model for Fractional Flow Reserve (FFR) Estimation" *Fractal and Fractional* 7, no. 5: 373.
https://doi.org/10.3390/fractalfract7050373