# Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator

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## Abstract

**:**

## 1. Introduction

#### 1.1. Review on Cardiovascular Conduction Process towards the Prosthetic Heart

#### 1.2. Importance of van der Pol Equation

#### 1.3. Why Hamiltonian Approach

#### 1.4. Background and Related Work

#### 1.5. Contributions and Organization

#### 1.6. General Objective of the Study

## 2. Modified Symmetric Oscillator Equation-Based Non-Linear Model of the Proposed Design

## 3. Hamiltonian Approach for Linear System

## 4. Hamiltonian Approach for Non-Linear System

## 5. Modeling of the Proposed Design

_{1}and k

_{2}represents the linear coefficient of spring, whereas ${\mathrm{k}}_{1}{}^{\mathrm{N}}$ and ${\mathrm{k}}_{2}{}^{\mathrm{N}}$ represents the non-linear coefficient of spring. d1 and d2 represent the displacement of mass M. Applying Hamiltonian H to the system $\hat{\mathrm{H}}$ can be written as

_{1}= ${\mathrm{B}}_{1}\mathrm{cos}\mathsf{\omega}\mathrm{t}$ and d

_{2}= $({\mathrm{B}}_{1}-\mathrm{B})\mathrm{cos}\mathsf{\omega}\mathrm{t}$ and then putting the value d

_{1}(t) and d2(t) in Equation (30), $\hat{\mathrm{H}}$ can be written as given below.

## 6. Generation of Action Potential and Coupling of Nodes

## 7. Muscle Dynamics Traditional Describing Function Method to Determine Nonlinearity

## 8. Simulation Result

## 9. Comparative Study of Our Proposed Design with Traditional Describing Function and Future Scope

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Plot of x1(t) and x2(t) when ${\mathrm{k}}_{1}{}^{\mathrm{N}}$ = 1, ${\mathrm{k}}_{2}{}^{\mathrm{N}}$ = 5.

**Figure 6.**Plot of x1(t) and x2(t) when ${\mathrm{k}}_{1}{}^{\mathrm{N}}$ = 3, ${\mathrm{k}}_{2}{}^{\mathrm{N}}$ = 7.

**Figure 7.**(

**a**) Represents a 2D phase portrait considering the different initial conditions. (

**b**) Represents a 3D phase portrait considering the different initial conditions.

An Author with Reference Number | Theme | Future Scope | Limitation |
---|---|---|---|

Yamada [19] | An implantable motor-driven artificial heart was tested. | The pump-based system tested on some animals may improve system efficiency. | The control analogy of this non-linear system is not illustrated. |

Love [20] | A mechanical, hydraulic circulation system was tested for developing an artificial heart system. | This mechanical circulation system generates an initial idea of the artificial heart. | The system and control dynamics is not mentioned. |

Kurita [21] | The whole operation is described as the development of an artificial heart. | Modification of the pump system provides development towards the artificial heart. | The heart model’s pumping operation is proposed, but the other operational parameters are not considered. For example, transportation delay, nonlinearity |

Konieczny [22] | The non-invasive pressure monitoring system in an artificial heart was proposed and analyzed. | The future scope is aimed at the development of non-invasive heart prostheses or assistive devices. | The pressure sensing technique for various conditions was not illustrated. |

Baldoni [23] | A magnetic heart valve sensing system was proposed for tracking heart valve prosthetic. | The monitoring system will be a step towards the development of the entire prosthetic heart monitoring system | The valve activity during monitoring is elaborated in this study, but the controlling analogy is not exact. |

Rosli [24] | A smart wireless heart monitoring system was developed that uses heartbeat data for alarming. | This system may recognize the specific heart problem and alert the medical person. | The parametric analysis on several critical factors like system stability, sensitivity, and reliability was not pointed out. |

Marom [25] | Three different Numerical models of artificial heart systems were developed in the aspect of flow control. | The various numerical model may develop the artificial model of the human heart. | The proposed design is incapable of generating a transfer function analogy. |

Shi [26] | The proposed artificial heart model was based on physical and numerical analysis. | The actual artificial heart model may be developed by electrical and mathematical analysis, comparing with a simulative approach. | Analysis of disease-related issues with cardiovascular system is depicted in this article. |

Pohlmann [27] | Some optimization techniques were realized in the artificial heart for optimal system performance. | The optimization technique drives towards functional improvement in the development of the artificial heart. | Optimization parameters of artificial heart modeling systems have not been considered. |

Symbol | Meaning | Unit |
---|---|---|

β | Damping constant | lbfs/inch |

$\mathsf{\sigma}$ | Constant of van der Pol model | NA |

x | Displacement | cm |

$\dot{\mathrm{x}}$ | Velocity | cm/s |

X | The amplitude of input sinusoidal signal for non-linear analysis | V |

M | The distance between nodes | mm |

Y | The output of the non-linear system | V |

M | Mass of cardiovascular muscle | g |

K | Spring constant | N/m |

B | Viscous drug | P |

N | Order of spring constant | NA |

K | The slope of non-linear section | NA |

${\mathsf{\delta}}_{\mathrm{ij}}$ | Kronecker delta | NA |

KE | Kinetic energy | J |

PE | Potential energy | J |

Px | Momentum | kg. m/s |

H | Hamiltonian operator | NA |

$\hat{\mathrm{H}}$(v) | Hamiltonian considering multiple inputs | NA |

L | Lagrangian operator | NA |

$\mathsf{\omega}$ | Angular frequency | rad/s |

T | Period | s |

$\mathsf{\Phi}$ | Phase shift | rad |

V | Describing function | NA |

${\mathrm{B}}_{\mathrm{n}}$ | Amplitude of oscillation | V |

Parameter | Values |
---|---|

${\mathrm{k}}_{1}{}^{\mathrm{N}}$ | 1,3 |

${\mathrm{k}}_{2}{}^{\mathrm{N}}$ | 5,7 |

M | 1 |

B | 10 |

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**MDPI and ACS Style**

Bhattacharjee, S.; Banerjee, A.; Rakshit, A.; Bhattacharyya, S.; Chowdhuri, S.; Sarkar, B.; Neogi, B.
Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator. *Symmetry* **2021**, *13*, 151.
https://doi.org/10.3390/sym13010151

**AMA Style**

Bhattacharjee S, Banerjee A, Rakshit A, Bhattacharyya S, Chowdhuri S, Sarkar B, Neogi B.
Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator. *Symmetry*. 2021; 13(1):151.
https://doi.org/10.3390/sym13010151

**Chicago/Turabian Style**

Bhattacharjee, Soumyendu, Aishwarya Banerjee, Amit Rakshit, Swapan Bhattacharyya, Swati Chowdhuri, Biswajit Sarkar, and Biswarup Neogi.
2021. "Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator" *Symmetry* 13, no. 1: 151.
https://doi.org/10.3390/sym13010151