Mathematical modelling and critical assessment of analytical solutions of forced-damped vibrations of the cardiovascular-implant system

A recent innovation in bioelectronic medicine is the use of implantable devices capable of harvesting biomechanical energy from cardiac motion. Such self-powered devices would facilitate cardiovascular functionality in patients with compromised hearts. This not only requires integrating bioelectronic medicine with cardiovascular physiology, but also a quantitative predictability of their functioning. We present a first attempt to establish a quantitative basis derived through biophysical considerations. Assuming cardiac functionality to be described using a spring-dashpot model, we present analytical solutions for different scenarios of physiological relevance. A key result is that the inverse lifetime lower than the natural frequency of the heart vibration leads to a rapid decrease in vibrational amplitudes of the implant as the cardiac cycle moves to the relaxation phase. When the inverse lifetime equals the natural frequency, vibrations persist to the largest extent and a substantial amount of energy can be harvested in a cardiac cycle via energy harvesting mechanisms (piezoelectric and triboelectric). Our analysis points to the critical role of the implant mass on variations in displacement during heart vibrations. Our theoretical predictions provide guidelines for developing next-generation biomedical devices with the heart as the <i>in vivo</i> source of energy harvesting.