In recent years, advances in ventricular assist devices (VAD) technology have allowed doctors to sustain and extend the lives of patients with cardiovascular disease with considerable success. Mechanical cardiac assistance in the form of VADs has become established as a therapeutic solution to congestive heart failure (CHF), which is currently the leading cause of death both in the United Kingdom and the United States.1,2 Since the traditional approach of heart transplantation is limited to helping approximately 10% of end-stage CHF patients, the ability to develop optimal mechanical solutions for ventricular assistance will help alleviate the growing problem. Various advantages, including lack of valves, compact size, and low power consumption, have made axial flow VADs the preferred approach for the next generation of blood pumps.
Mild CHF often reduces the ability of vital abdominal organs, such as liver and kidneys, causing further patient deterioration. At the same time, existent VAD problems and the invasiveness of heart surgery limit mechanical assistance to only the severely ill patients. Implementing an insertable micro-axial blood pump placed in the descending aorta to augment renal blood flow could minimize surgical complications and extend VAD use to less severe CHF patients. The proposed VAD cooperates in series with the natural heart which remains active and functional in early-stage CHF. Further research is needed due to the particular location and working criteria of the pump, as challenges arise in large air-gap motor design, mechanical structure of the impeller, sophistication of control algorithm, as well as hemodynamic considerations beyond the scope of this discussion. In this paper, a mathematical model is presented for better understanding of the interactions between the VAD and the human cardiovascular system (HCS), and can be used as an accurate and adaptive model for initial pump design.
To date, numerous mathematical HCS models of differing complexities have been developed to incorporate physiological parameters and produce simulated hemodynamic responses of various pathological conditions.3–5 These computational models, which are represented by analogous electrical elements as lumped-parameter circuits, offer valuable insight into the complicated HCS dynamics for specific purposes.6–8 The simplified circuit analog of the HCS, where the entire systemic circulation can be lumped as one resistance element ]the total peripheral resistance (TPR)[ has been recognized as a useful approach to the development of VAD physiological control algorithms and assignment of pump parameters in the recent literature.9,10 Xu and Fu11 have demonstrated a regulated circulatory system that interacts with mathematical representations of the electric motor and blood pump. Wu et al.12,13 have proposed an adaptive physiological controller for a permanent VAD coupled with a nonlinear lumped-parameter model of the HCS. Although these computational models are effective for conventional VAD design, a more detailed model is needed to assess regional hemodynamic effects, such as renal or cerebral perfusion, as a result of pump operation. And while pump models based on empirical data are available, usage is limited only to their respective devices. Furthermore, minimal literature exists in the area of VADs working in series with the heart, which requires investigation of the pump’s upstream and downstream conditions (i.e., pressure and flow rate in the descending aorta for our application).
The present model has been extended from the classical closed-loop lumped-parameter HCS model, PHYSBE,14 implemented in MATLAB/Simulink (The MathWorks, Natick, MA). The systemic circulation and descending aorta were further divided to facilitate a comprehensive model that avoids the complexity of a full-scaled model while preserving the essential characteristics of major cardiovascular subsystems. Classical turbomachine theory was used to construct the general characteristic curve for the axial-flow impeller. This study provides a mathematical model of the HCS coupled with a VAD to analyze the hemodynamic responses of local regions, allowing investigation of the general governing equations for initial pump design.
The left and right hearts are modeled as variable capacitors with a modified pressure-volume relationship based on Wu et al.12 and given by the following equation:
where P and V represent ventricular pressure and volume; a, b are diastolic elasticity coefficients; V0 is the zero-pressure volume of the ventricle; Emax is the peak ventricular elasticity during systole; tp, ts, and tT indicate peak of cardiac contraction, period of systole, and one cardiac cycle, respectively. Numerical values for above variables under healthy and CHF are presented in Table 1.
Elv/Erv: Emax of left and right ventricle; V0_lv: V0 of left ventricle (LV); TPR: mean aortic pressure divided by cardiac output (CO).
The model’s four heart valves are modeled as diodes with a delay introduced in the closing time to gain an accurate physiological correspondence. Due to the specific location of the proposed VAD, a detailed descending aortic model composed of three subsections (thoracic, descending, abdominal aorta) with corresponding resistances and compliances was developed in the CAM to yield an accurate system response after insertion.
Parameter assignments for the CAM are based on published values wherever possible.4,15 Remaining unknowns including parameters in the subsections of aorta and regional systemic circulatory branches were estimated and adjusted accordingly. The parameter sets were then optimized through an exhaustive numerical search and chosen to fit the average healthy or CHF patient.
Analytical Model of the Axial Flow Impeller
The proposed axial-flow pump is positioned in the descending aorta above the renal arterial junction to increase renal perfusion (Figure 3). Insertion via the femoral artery is intended to minimize surgical complications; as a result of this constraint, the radii of the blade tip and hub are limited to 10 mm and 5 mm, respectively. Unlike existing VADs, the proposed intra-aortic pump is designed to work in series with the weakened LV and partially offload ventricular duty without replacing the heart in early stage CHF. Since the impeller operates under a highly pulsatile flow condition, the necessity of synchrony between the heart and device is presumed. The VAD is driven by a brushless DC (BLDC) motor, but a practical control strategy has yet to be designed. For the present case, the pump is driven and controlled by a pulsatile speed profile.
The fundamental turbomachinery formulae widely used in conventional rotary machines was applied to define the basic blade geometry, by choosing optimized inlet and outlet angles for required flow rates (Q) and total pressure rises (ΔP)16; dimensionless parameters are often used to obtain estimations of initial design characteristics.9,17 For pumps working with an incompressible fluid, Q and ΔP are combined into two nondimensional variables, the head coefficient (ψ) and the flow coefficient (φ), to characterize the pump performance:
where ΔP denotes the impeller pressure difference between impeller upstream and downstream, ρ, U, and vx represent the blood density, blade circumferential speed, and axial flow velocity, respectively. The steady-state relationship between ψ and φ of an axial-flow impeller blade (with no inlet prerotation) can be expressed using the relative outlet flow velocity angle (β2) and loss coefficient (CD)18 through the following:
where σ is solidity of the blade (which equals chord length l divided by span s) and βm is mean relative flow velocity angle. The loss coefficient CD is made up from1 profile loss,2 annular loss,3 secondary loss, and4 leakage loss, given by19:
where CDP, CDa, CDS, CDk represent the four losses above, respectively, and are incorporated into the analytical impeller model. Losses are defined by classic aerodynamic correlations20 to give reasonable predictions of impeller performance.
Due to the pulsatile blood flow in HCS, the transient, unsteady behavior of the pump must be considered. The steady-state pressure head, ΔP, is adjusted by a time-derivative term to yield a more realistic impeller model. Applying the unsteady Bernoulli’s equation to the flow in the impeller and a first approximation of the difference of the velocity potential21, the dynamic pressure head across the pump ΔPd can be derived as:
The steady-state characteristic equation of an axial-flow VAD (Equation 3) is defined with user-specified blade geometry through inlet and outlet angles, impeller length and diameter, blood viscosity, and number of blades. A four-blade single-stage micro-axial impeller with inlet angle, outlet angle, blade-tip radius, blade height, and chord length of 21°, 32°, 1 mm, 0.5 mm, and 3 mm, respectively, was implemented in the CAM for simulation. The steady-state pressure head ΔP is calculated using instant flow velocity and rotational speed, and modified by a dynamic term proportional to the time derivative of inflow velocity (Equation 5). To minimize calculation time, a reference table of steady-state pressure heads generated from the characteristic equation (Equation 3) is built into the pump model and coupled with the CAM.
The analytical impeller model is driven by a simple periodic speed profile only to demonstrate the system’s capability. Inlet flow rate and pressure output from the CAM along with controlled rotational speed input are fed into the impeller model to generate the downstream pressure. The outlet flow rate and pressure are then fed back into to the CAM which updates the entire system at the next iteration.
Transient Hemodynamic Response
Simulations were performed at a heart rate of 75 bpm under different physiological conditions. By introducing blood inertia, phase shifts and dicrotic notches of the pressure waveform along the aorta were simulated. The resulting waveforms show a more physiological response as compared with the conventional lumped RC model (Figure 4).
Figure 5A illustrates the healthy and CHF LV pressure- volume loops produced by the CAM. Mild left ventricular dysfunction is modeled by selecting proper values for Emax, alv, and blv to produce a 45% decrease in stroke volume and a rise to 19 mm Hg in EDLVP. The CO of healthy and CHF cases are shown in Figure 5, B and C, respectively; the average flow rates (Table 2) correspond with the values reported in current literature. Hemodynamic Parameter Estimations with VAD.
The pump characteristic curve defined by Equation 3 is illustrated in Figure 6A for selected rotational speeds. The proposed axial-flow VAD was driven by a sinusoidal speed profile (Figure 6B) which is synchronous with the natural heart beat. For demonstrating the feasibility of this device, the average flow and pressure estimations were sufficient for an initial design point. The simulated results of healthy, CHF, and postinsertion cases are averaged over the steady state and summarized in Table 2.
A widely known difficulty in developing lumped models is the constant assignment. Although most parameters are chosen based on anatomical data and general physiological calculations, it is a tedious process to optimize the values in lumped models. If the computational time is lengthy (taking 10 seconds to run a 40-second simulation), interpreting the outcome of each individual parameter would be taxing. Therefore, direct experimental data could help set the parameters for such a system.
Figure 7 shows the pressure head across the impeller (top) and inlet flow (bottom) during the operating cycle when rotational speed was set between 2,400 and 3,200 rpm. Despite the simple control algorithm, the results provide valuable information in dynamic interaction between the VAD and circulatory system. Unsteadiness of pump (such as stall or surge) can happen with improper rotational speed selection, which could be fatal in this specific application. Simulation results show unsteady behavior appearing as pressure fluctuations in the descending aorta during diastole when rotational speed is too high. The authors acknowledge the importance of real-time control of the impeller speed to maintain a proper pressure along the aorta during the each cardiac cycle; further research must be done to ensure such stable pump operation.
The presented model, CAM, has extended the previous lumped-parameter HCS model that is generally used in VAD research and provides insight into global as well as regional hemodynamic interactions. The dynamic responses are physiologically reasonable and mean values of estimated parameters agree with values in literature. The major difficulties, as reported in previous research on circuit equivalent HCS models, were to fit the physiological parameters and acquire the regional hemodynamic profiles. Nevertheless, the simulation can help VAD designers make initial estimations of the performance of a blood pump, especially for applications where there is a particular interest in regional hemodynamics. The results can also reveal priorities for device optimization before costly, time-consuming computational fluid dynamics, in vitro, and in vivo experiments. Additionally, the model could be used to assess faulty conditions during VAD operation such as ventricular or aortic suction.
Although the 2-D turbomachinery equations may underestimate the loss that occurs in the gap between vessel and impeller tip as well as other hemodynamic effects, the analytical axial VAD controlled by a periodic speed profile demonstrated the feasibility of a descending-aortic device concept to augment renal perfusion. The VAD model can be readily implemented in different locations to simulate various other possibilities.
As future work, a physical mock circulatory system will be used to test the impeller and uniquely designed motor drive system. A sophisticated control strategy is crucial and will be developed using the presented system to generate better pump performance. The CAM implemented in the widely used MATLAB environment eases the future integration of entire analytical systems including the impeller, electric motor, and HCS for early stage design of innovative axial-flow VADs.
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