## 2 Materials and Methods  
Computer simulations were performed using a cardiovascular system model which includes heart chambers, heart valves, aorta, systemic arterioles and veins, pulmonary arteries, arterioles and veins. LVAD support was simulated by implementing numerical models which simulate pressure head and flow rate (H-Q) relations in HVAD, HM2 and HM3 devices. The cardiovascular system model simulates pressure, volume and diameter in the heart chambers; flow rate through the heart valves; and pressure and flow rate in the systemic and pulmonary circulations. In this model, left-ventricular pressure (\(p_{lv}\)) is described using the active and passive contraction components (\(p_{lv,a}\), \(p_{lv,p}\)). Ventricular active-pressure component (\(p_{lv,a}\)) is described using the systolic ventricular elastance (\(E_{es,lv}\)), ventricular and zero-pressure volume (\(V_{lv}\), \(V_{lv,0}\)) and the activation function (\(f_{act,lv}\)). Ventricular passive-pressure component (\(p_{lv,p}\)) is modelled using an exponential relationship, including volume (\(V_{lv}\)) and additional parameters (\(A\), \(B\)). \[p_{lv}=p_{lv,a}+p_{lv,p} \tag{1}\]  
\[p_{lv,a}=E_{es,lv}(V_{lv}-V_{lv,0})f_{act,lv}(t) \tag{2}\]  
\[p_{lv,p}=A\left[e^{(BV_{lv})}-1\right] \tag{3}\]  
Left-ventricular volume (\(V_{lv}\)) is described using the left-ventricular radius (\(r_{lv}\)), long axis length (\(l_{lv}\)) and an additional coefficient (\(K_{lv}\)), which includes effects of the contraction in the long axis and scales the proportion between the left-ventricular radius and volume over a cardiac cycle. Change in the left-ventricular radius (\(r_{lv}\)) is described utilising the flow rates through the aortic and mitral valves (\(Q_{av}\), \(Q_{mv}\)), left-ventricular volume, long axis length (\(l_{lv}\)) and the coefficient \(K_{lv}\). \[V_{lv}=\frac{2}{3}\pi K_{lv}r_{lv}^{2}l_{lv} \tag{4}\]  
\[\frac{dr_{lv}}{dt}=\frac{3(Q_{mv}-Q_{av})}{4\pi K_{lv}l_{lv}}\left(\frac{3V_{lv }}{2\pi K_{lv}l_{lv}}\right)^{-1/2} \tag{5}\]Right and left ventricles are modelled in the same way using different parameter values. The left atrial pressure (\(p_{la}\)) and volume (\(V_{la}\)) relationship is described using elastance (\(E_{la}\)). \[p_{la}=E_{la}(t)(V_{la}-V_{la,0}) \tag{6}\]  
Left atrial volume (\(V_{la}\)) is described using the left atrial radius (\(r_{la}\)), long axis length (\(l_{la}\)) and an additional coefficient (\(K_{la}\)). Change in the left atrial radius (\(r_{la}\)) is described utilising the flow rates through the mitral valve and pulmonary vein (\(Q_{mv}\), \(Q_{vp}\)), left atrial volume, long axis length (\(l_{la}\)) and the coefficient \(K_{la}\). \[V_{la}=\frac{2}{3}\pi K_{la}r_{la}^{2}l_{la} \tag{7}\]  
\[\frac{dr_{la}}{dt}=\frac{3(Q_{vp}-Q_{mv})}{4\pi K_{la}l_{la}}\bigg{(}\frac{3V_{ la}}{2\pi K_{la}l_{la}}\bigg{)}^{-1/2} \tag{8}\]  
Right and left atria are modelled in the same way using different parameter values. Heart valves are modelled as ideal diodes using pressure across a valve and the characteristic resistance. Flow rate through the mitral valve (\(Q_{mv}\)) is given below. \[Q_{mv}=\left\{ \begin{array}{cc}\frac{p_{la}-p_{la}}{R_{mv}}&p_{la}>p_{lv}\\ 0&p_{la}<p_{lv}\end{array} \right. \tag{9}\]  
The other heart valves are modelled in a similar way using different parameter values. The circulatory system includes aorta, systemic arterioles, systemic veins, pulmonary artery, pulmonary arterioles and pulmonary veins. Blood flow in the circulatory-system model is described using a lumped-parameter model, which includes electrical analogues for resistance (\(R\)), compliance (\(C\)) and inertia (\(L\)). The aortic blood pressure (\(p_{ao}\)) and flow rate signals (\(Q_{ao}\)) are given below.