Consider the simplified model of the coupled LVAD-cardiovascular system as shown
ID: 2082859 • Letter: C
Question
Consider the simplified model of the coupled LVAD-cardiovascular system as shown below: Assume that the pump (LVAD) pressure (head) gain is given by H_p (t) = A sin (omega t) + B, to: a) Derive the system of seven (7) switch-state equations with the corresponding state matrices [-A (t)] and right-hand side vectors [b (t)}for the four (4) different valve states (1: closed-closed. 2: closed-open. 3: open-closed. 4: open-open) b) Numerically approximate the solution of the system with the theta (Delta t^4) Runge-Kutta method using any mathematical software (MathCAD. MATLAB. Mathematica. Maple, etc.) or any high-level programming language (C. C++, Fortran. Java, etc.) using the typical adult RLC parameters in the circuit (and L_SYS = 0.001 mmHg/ml/s^2, A = 30 mmHg. B = 50 mmHg. and the pump frequency omega as the same as the frequency of the heart cycle). Let the heart rate be 75 BPM and setup the time step to Delta t = 0.0015 to generate the numerical approximation over 5 heart cycles with E_max = 0.7 mmHg/ml and E_min = 0.06 mmHg/ml. (Use the following initial conditions: P_LA = 7 mmHg, P_LV = 7 mmHg. P_AO = 60 mmHg, Q_AO = 0 mt/s. P_SYS = 60 mmHg. Q_p = 0 ml/s. Q_SYS = 0 ml/s) c) Plot the four (4) state pressures (P_LA. P_LV. P_AO. P_SYS) along with the pump pressure (H_P) over the first five (5) heart cycles. d) Plot the three (3) state flowrates (Q_AO. Q_P. Q_SYS) over the first five (5) heart cycles. c) Calculate the average pump flowrate (QP). aortic flowrate (Q_AO), the Cardiac Ejection (Q_cardiac). and the systemic flowrate (Q_SYS) in l/min over the fifth (5^th) cardiac cycle.Explanation / Answer
The baseline geometry of HTA includes an inducer at the inlet, a diffuser at the outlet, and a rotating impeller. These components were expressed using eleven design parameters. The design parameters were optimized using genetic algorithms in order to increase pressure rise at the outlet of the pump and reduce the wall shear stresses on the surface of the components for 6L/min flow rate through the pump. The optimized geometries were assessed employing Computational Fluid Dynamics (CFD) analyses. The geometry generates the highest pressure rise at the outlet of the pump and lower wall shear stress on the components was selected as HTA CF-LVAD. The detailed information about the shape optimization and CFD analyses can be found in [17, 22].
A model was chosen to estimate the pressure head of HTA using pump flow rate and operating speed and the coefficients were estimated using the results of the CFD analyses to describe HTA. This model is given as follows:
Here,is the pressure difference across HTA andandare the flow rate through HTA and the operating speed of HTA, respectively.andare the estimated coefficients. The least squares approximation was used to estimateand. Estimated values ofandwere 0.2380mmHgmL1 and 0.0032mmHgs2 with a root mean square error of 1.41mmHg. The detailed information about the parameter estimation is give
n in [23]. Data from the CFD analyses and the estimated pump performance curves are given in Figure 1.