## 2 Materials and Methods
### Continuous Flow Left Ventricular Assist Device (CF-LVAD) Support  
To simulate CF-LVAD support, a model that estimates the pressure difference across an axial CF-LVAD (\(\Delta\)pcf-lvad), considering operating speed of the pump (\(\omega_{\text{CF-LVAD}}\)), flow rate (QCF-lvad) and change of the flow rate through the pump (dQCF-lvad/dt) [36], was integrated into the cardiovascular system model.  
\[\Delta\text{p}_{\text{CF-LVAD}}=\text{K}\omega_{\text{CF-LVAD}}^{2}-\text{R} _{\text{CF-LVAD}}\text{Q}_{\text{CF-LVAD}}-\text{L}_{\text{CF-LVAD}}\frac{ \text{d}\text{Q}_{\text{CF-LVAD}}}{\text{dt}}, \tag{20}\]  
\[\text{R}_{\text{CF-LVAD}}=\text{k}_{1}\text{Q}_{\text{CF-LVAD}}+\text{k}_{2}, \tag{21}\]  
In the equations above, LCF-lvad and RCF-lvad are the inertance and resistance effects in the pump, respectively. K, k\({}_{1}\) and k\({}_{2}\) are the estimated parameters [36]. The CF-LVAD was operated at a 10,500 rpm rotation speed in the cardiovascular system models simulating the heart pump support. The CF-LVAD model was implemented into the cardiovascular system model by modifying the equations which describe change of the left ventricular radius (Equation (5)) and aortic pressure (Equation (19)) as below.  
\[\frac{\text{d}\text{r}_{\text{lv}}}{\text{dt}}=\frac{3(\text{Q}_{\text{mv}}- \text{Q}_{\text{av}}-\text{Q}_{\text{CF-LVAD}})}{4\pi\text{K}_{\text{lv}} \text{l}_{\text{lv}}}\Bigg{(}\frac{3\text{V}_{\text{lv}}}{2\pi\text{K}_{\text{lv }}\text{l}_{\text{lv}}}\Bigg{)}^{-1/2}, \tag{22}\]\[\frac{\mathrm{dp_{ao}}}{\mathrm{dt}}=\frac{\mathrm{Q_{av}+Q_{CF-LVAD}-Q_{ao}}}{ \mathrm{C_{ao}}}. \tag{23}\]  
Systemic arteriolar resistance was 1.25 mmHg/mL for the cardiovascular system model simulating heart failure as it increases to maintain perfusion levels in the body for a failing heart [37]. It was reduced to a healthy value, 0.75 mmHg/mL, for CF-LVAD support in the simulations, because mechanical circulatory support restores the blood flow, and pressure levels in circulatory system baroreflex control is expected to reduce the peripheral systemic resistance with increasing arterial pressure [38].  
The electric-analogue of the heart chambers and the circulatory loop used to simulate normal SR and AF with unimodal and bimodal RR interval distributions for heart failure and CF-LVAD support is given in Figure 2. The list of abbreviations and the parameter values used in the circulatory loop and cardiac functions is given in the Appendix A section.  
The simulations were performed using MATLAB Simulink R2017a. The set of equations was solved using the ode15 s solver. The maximum step size was 1e-3 s and the relative tolerance was set to 1e-3. All of the cardiovascular system models were run at a 40 s simulation time at a 75 bpm constant heart rate, to obtain a periodic steady state solution, before the RR intervals were used as temporal ranges for the heartbeats.  
Figure 2: The electric-analogue of the heart chambers and circulatory loop used to simulate normal sinus rhythm and atrial fibrillation with unimodal and bimodal RR interval distributions for heart failure and Continuous Flow Left Ventricular Assist Device (CF-LVAD) support.