Analytical solution to Windkessel models using piecewise linear aortic flow waveform.

Area of Science:

• Physiology
• Biomedical Engineering
• Mathematical Modeling

Background:

• Windkessel models are crucial for analyzing arterial pressure-flow dynamics in cardiovascular research and education.
• Current time-domain analyses often rely on numerical methods or frequency-domain approaches, limiting accessibility and interpretability.
• Existing methods can be computationally intensive and may obscure the underlying physiological relationships.

Purpose of the Study:

• To derive time-domain analytical solutions for two-, three-, and four-element Windkessel models.
• To provide explicit, exact, and easily understood mathematical expressions for arterial pressure.
• To offer an alternative to numerical integration and frequency-domain analysis for Windkessel models.

Main Methods:

• Solving the first-order, non-homogeneous, linear differential equations governing the Windkessel models analytically.
• Utilizing a piecewise linear function to accurately approximate typical aortic flow waveforms.
• Obtaining closed-form expressions for arterial pressure under transient and steady-state periodic conditions.

Main Results:

• Derivation of closed-form analytical solutions for arterial pressure in multi-element Windkessel models.
• Demonstration of the ability to accurately represent arterial pressure in both transient and steady-state regimes.
• The derived solutions are explicit, exact, and mathematically straightforward.

Conclusions:

• The developed analytical solutions provide a significant advancement for time-domain analysis of Windkessel models.
• These solutions simplify the study of arterial pressure-flow relationships, enhancing educational and research applications.
• The approach eliminates the need for complex numerical solvers and Fourier analysis, offering a more direct understanding of model behavior.