Because of the limited number of donor hearts available each year, an alternative treatment for cardiac patients is to use the mechanical circulatory support device.1 The miniature axial blood pump has been given more attention for the past 2 decades as a result of its small size,2 simplified implantation provided by its tubular configuration,3 lower power consumption requirements, and potential for low-cost manufacturing. Conduit-free pumps are novel miniature axial blood pumps, including the dynamic aortic valve (DAV),4 the aortic valve pump,5 and the functionally total artificial heart (artery pumps).6 All of these can deliver blood flow directly from left ventricle to aorta similar to the natural heart without connecting conduits or bypass pathway, thus eliminating the most dangerous sites of thrombosis in traditional blood pumps.6 Unfortunately, when a conduit-free pump is implanted at the place of aortic valve, the aortic valve may be damaged to some degree, because of the transplantation or the long-duration pressure, which forces the aortic valve to open throughout the entire cardiac cycle. Therefore, the latest conduit-free pump, an intra-aorta pump (Figure 1),7 was developed autonomously by the artificial heart research group at Beijing University of Technology. The pump can be implanted between the radix aortae and aortic arch to avoid damage to the aortic valve. The pump includes a rotating impeller with blades to impart kinetic energy to the blood, a diffuser with helically curved blades, a pair of bearings, and housing. The pump has a length of 40 mm, the largest diameter of 20 mm, and a weight of 40 g. It is driven by an extracorporeal dynamic system that is placed in vitro. The rotational speed of the impeller can be regulated by the dynamic system.
Essential analyses of the fluid dynamic characteristics of blood pumps, involving numerical and experimental studies, are performed worldwide for refining the blood pump structure. In 2000, Anderson et al.8 established a model for the CFVAD3. He used the model to calculate the fluid dynamic characteristics of the left ventricular assist device (LVAD) by using computational fluid dynamics (CFD) for refining the CFVAD3 structure and predicting the blood damage, and it achieved a good performance. However, using CFD needs a couple of days to calculate; hence, this method does not fit for the real-time calculation in clinical application. To overcome this problem, researchers proposed lumped parameter models to estimate the blood flow and differential pressure. In 2002, Giridharan et al.9 reported a lumped parameter model of the VAD. The load torque of the VAD is defined as a cubic function of the blood flow and rotational speed. The blood flow and pressure head across the VAD are calculated from an empirical formula whose parameters are determined experimentally. In 2005, Boston and coworkers10 proposed a nonlinear time-varying model. The model consists of a lumped parameter model of the heart and an empirical formula of the Nimbus pump, which is a function of the pressure head, blood flow, and the rotational speed. To simulate the phenomenon of suction, a time varying, nonlinear, pressure-dependent resistor is added to the model. However, the resistor has no physical significance. Because the resistor is a nonlinear function of the pressure, the model is too complex for calculating the blood flow and pressure. Moreover, the structure and parameters of the model are determined experimentally, i.e., the structure and values of parameters will be disturbed by measurement noise. Xia and Bai11 proposed a nonlinear lumped parameter model for the DAV and the circulatory system in 2005. The model is used to calculate the blood flow and pressure head after the DAV being implanted at the place of the aortic valve, which achieved a good performance. However, the model includes parameters of the circulatory system, and it is too complex to calculate real time. Yu et al.12 in 2008 reported a nonlinear circuit model for the percutaneous ventricular assist system. The model consists of the lumped parameter models of the radix aortae, aortic arch, and an empirical formula of the blood flow and pressure head. The lumped parameter models especially achieve good performance on simulating the blood flow and pressure head. However, the relationship of the blood flow and pressure head are defined by the empirical formulas, where the physical significances of the parameters are not clear; hence, it is hard to extend these formulas to other blood pumps. In addition, the LVAD in these models are all represented as voltage source, and they cannot simulate the status of the pump when suction occurs. To solve this problem, pressure-controlled resistors have been used, but the resistors have no physical significance in the actual LVAD system.
In this article, a nonlinear model of the intra-aorta pump is established, which does not consist of the parameters of the circulatory system. The model is based on the development of a nonlinear lumped parameters model for the intra-aorta pump in which the impeller of the pump is modeled as a speed-controlled current source, an internal resistor is used to simulate the resistance character of the radial clearance of pump, and an inductance is used to model the inertia of blood that passes through the pump. The model represents the differential pressure over the pump as a linear function of blood flow in it. It can simulate overall status of the pump from suction to pulmonary congestion. Each part of the model has clear physical significance that is helpful for extending the model to other blood pumps. To test the accuracy of the model and prediction method, benchtop experiments are conducted on an in vitro mock loop.
Materials and Methods
Description and Analysis of Systemic Circulation Model
The blood flow and pressure head in the vessels are governed by Poiseuille's law. That is
where P represents the pressure head across the vessels, Q the blood flow in the vessels, and f(·) a continuously differentiable nonlinear function that satisfies f(0) = 0. However, according to recent research,11,13 the pressure head through the pump decreases with increase in blood flow across the pump. Hence, when the blood flow across the intra-aorta pump is zero, there exists a function C(ω) that satisfies Equation 2:
where ω is rotational speed of the pump. C(ω) and R0(ω) are polynomial coefficients that are determined experimentally by fitting expression. From Equation 2, the intra-aorta pump circulatory system is modeled with an equivalent electric circuit (Figure 2). In this approach, the pressure is modeled as the voltage and the blood flow by the current. The model of intra-aorta pump is completely different from other lumped parameter models. The models proposed by other groups represent the LVAD as voltage sources.12,14 For instance, Simaan et al.14 proposed a lumped parameter model of LVAD in 2009. The LVAD is modeled as a voltage source that is a nonlinear function of rotational speed and blood flow through the pump. However, when the phenomenon of ventricular collapse or pulmonary congestion occurs, this model cannot simulate the status of the pump (such as the blood flow through the pump and the pressure at the inlet of the pump). To solve this problem, a time varying, nonlinear, pressure-dependent resistor is added to the model to generate negative pressure at the inlet of the pump. However, this resistor has no actual physical significance. The only role of the resistor is to simulate the phenomenon of suction and pulmonary congestion in mathematics. On the contrary, the intra-aorta pump is modeled as a speed-controlled current source with an internal resistor and internal inductance. The internal resistor represents resistance character of radial clearance of the pump (Figure 3), and the internal inductance represents the inertia of the blood in the radial clearance of the pump. This model can simulate the phenomenon of suction and pulmonary congestion without other parts that have no physical significance. For example, when pulmonary congestion occurs, the quantity of blood flow through the pump is more than the one that the pump can deliver, i.e., QPO > QPS(ω). The error of the blood flow is QP1, which passes through the pump from the radial clearance. Because the QP1 that passes through the internal resistor and inductance, the pressure at the inlet of the pump is greater than the one at the outlet of the pump. The phenomenon is consistent with clinical records. It is seen that the model presented in this article has simpler structure than the one proposed previously, and each part of the model has clear physical significance.
According to Figure 2, the pressure head is a function of the rotational speed and the blood flow, which satisfies Equations 3 and 4:
where the blood flow of QPO is the output blood flow of the pump, QPS(ω) is the blood flow that is delivered by the kinetic energy from the impeller, which depends on the rotational speed of the impeller, and RPS(ω) denotes the resistance character of the radial clearance of the intra-aorta pump, which is a function of rotational speed. LP represents the inertia of the blood in the radial clearance of the pump,13 which is defined by Equation 4. l is the length of the intra-aorta pumps, rim is the diameter of the impeller, rho denotes the internal diameter of the housing, and ñ is the density of the blood. In Equation 3, there are no parameters of circulatory system, and it is only the function of rotational speed and blood flow through the pump.
Because the model is focused on the steady-state blood flow and pressure head, the derivative part of Equation 3 is small enough to be ignored. Therefore, the pressure–flow rate function is simplified as Equation 5:
where RPS(ω)QPS(ω) and RPS(ω) are defined as polynomial coefficients C and R0 in Equation 6. Hence, Equation 5 becomes the same form as Equation 2:
To test the model, benchtop experiments were conducted to derive data for identifying the model parameters and for testing the accuracy of model in predicting the differential pressure over the pump by changing the pump speed, blood flow, and peripheral resistance. The experiment setup consists of an in vitro mock loop shown in Figure 4, which includes a preload chamber modeling the left ventricular pressure; a φ 20 mm silicone tube mimicking the radix aortae; an intra-aorta pump that delivers the test fluid from the radix aortae to the aortic arch directly; a φ 20 mm silicon tube mimicking the aortic arch; a ultrasonic flow probe (Ultrasonic Pulsed Doppler Blood Flow System, Shanghai Alcott Biotech Co., Ltd, ALC-BFS) is placed after the pump outlet to measure blood flow through the pump; two pressure transducers are placed in the middle of the radix aortae and the aortic arch to measure the pressure at the inlet and the outlet of the intra-aorta pump; the tubing clamp can be adjusted manually to achieve a desired blood flow or differential pressure. The test fluid is 33% glycerin water (by volume) to simulate the viscosity of the blood at 37°C.
Experiments to Identify the Systemic Circulation Model Parameters
The above setup is first used to identify the internal resistor of the intra-aorta pump RPS(ω). As shown in Figure 5, the rotational speed is adjusted from 3,960 to 7,860 rpm with a 300 rpm increment; at each speed set point, the clamp was varied to adjust the blood flow from 2 to 7 L/min at a 1 L/min increment, and the pressure head was recorded at each blood flow set point. The results are illustrated in Figure 6, where the horizontal coordinate is the blood flow, i.e., from 2 to 7 L/min, and the vertical coordinate is the pressure head, which was from 0 to 180 mm Hg. Figure 5 depicts that the pressure head attributes approximate linear relationship with the blood flow; the increments of intercept increase when the rotational speed increases. RPS(ω) is computed as the slope of each line (Table 1). Using the least squares fit, RPS(ω) is represented by Equation 7:
To identify the polynomial coefficient RPS(ω)QPS(ω), similar experiments were performed. QPO was maintained at 0 L/min; the rotational speed was adjusted from 3,960 to 7,860 rpm at a 300 rpm increment; the pressure head was recorded at each speed set point; the data derived from these experiments are listed in Table 2.
According to the results of the experiments, RPS(ω)QPS(ω) is considered as a quadratic function of rotational speed. Using least squares fit, RPS(ω)QPS(ω) is expressed by Equation 8:
Substituting Equations 7 and 8 into Equation 5 yielded Equation 9:
where QPO represents the blood flow of the pump (in L/min); PP is the pressure head of the pump (mm Hg); ω denotes the rotational speed (in R/s); ωlimit is the threshold speed (in R/s); and g(·) denotes the step function that was denoted by Equation 10.
Prediction of the Pressure Head and Flow Rate by Applying the Model
To test the accuracy of the prediction method, other experiments are conducted. The intra-aorta pump is operated at the blood flow from 2 to 7 L/min with a 1 L/min increment; at each given flow rate, the clamp and rotational speed of the pump are adjusted to get the desired pressure head that is from 20 to 160 mm Hg, with a 10 mm Hg increment. The pressure head is recorded to compare with the predicted differential pressure obtained from the model by implementing Equation 9. The maximum error between the predicted pressure head and the measured one is calculated by implementing Equation 11:
where Emax is the maximum error; max(·) denotes the function that returns the maximum element, Pim represents the ith measurement pressure head; and PiP represents the ith predicted pressure head. Figure 7 shows plots of the predicted pressure head versus the measurement value over the entire range of mean aortic pressure (MAP). The horizontal coordinate is the measured pressure head, and the vertical ordinate is the predicted pressure head. Both the horizontal coordinate and the vertical coordinate are from 0 to 200 mm Hg. Figure 7, a–d presents the relationships of them when the blood flow is maintained at 3–6 L/min, respectively. We note that the slope of these lines is very close to 1, implying a very good prediction of the model to experimental data. Based on Equation 11, the maximum error is <5% relative to the measured data.
The intra-aorta pump is a novel LVAD with a simple structure. Because the blood pump has no percutaneous wires, it is difficult to derive the blood flow and pressure head signal directly. Therefore, a nonlinear lumped parameter model of the intra-aorta pump was proposed to calculate the blood flow and the pressure head on line. Unlike other lumped parameter models of the LVAD, in this article, the intra-aorta pump is modeled as a speed-controlled current source with an internal resistor and inductance. It has advantages over other models. On the one hand, compared with other models that use voltage source to represent LVAD, the model in this article can simulate overall status of the pump from suction to pulmonary congestion without any part that has no clear physical significance. The internal resistor represents the resistance character of the radial clearance. The internal inductance models the inertia of the blood in the radial clearance. That is, the function of the resistor and inductance can be determined when the structure of the pump is designed by using CFD. In addition, the experiments are used to test the accuracy of the model and to meliorate the error between design and actual work. In this way, the numerical simulation, lumped parameter model, and experiment can be combined to design LVAD. The lumped parameter model is used to establish the structure of the model; the numerical simulation is used to calculate the parameters, and experiment is used to test and meliorate the model. On the other hand, the model does not consist of the parameters of the circulatory system, which is helpful for extending the model to other blood pumps. Second, the model reported in this article is established based on both the analyses of hemodynamic and experiment data. The analysis of hemodynamic data is used to determine the structure of model, which consists of the order of variables and the number of parameters. The experiment data is only used to determine the value of parameters. If the pump has the similar H-Q relationship to that of intra-aorta pump, it only needs to use the experiment data to calculate the value of parameters. For other LVADs whose H-Q relationships are very different from that of intra-aorta pump, the model presented in this article cannot be used directly. The structure of the model needs to be modified to match the LVADs.
The outputs of static model in this article are the average value of hemodynamic parameters, such as the differential pressure over the pump and the flow rate through the pump. For example, the differential pressure PP in Equation 5 represents the average differential pressure over the pump. These average values of hemodynamic parameters have physiological significance. They can represent the status of circulating system and pump. In practical work, the average value of hemodynamic parameters has been used in various LVAD control system as control objects.15,16
The pressure head prediction model is derived under the steady-state blood flow condition on the in vitro mock loop, and the test fluid in the circulating system was not blood. Hence, the model does not include the effect of the flow pulsatility introduced by the native heart. Because the pressure–flow rate function focuses on the steady state of the pressure head and blood flow, the inertia of the blood and pump are ignored to simplify computation. Therefore, the model is not fit to predict the instantaneous state of the circulating system.17 To overcome this shortage, a pulsatile pressure–flow rate function will be achieved in the future by adding a mathematic model of the nature heart and the inertia of the blood flow.
A nonlinear lumped parameter model of the intra-aorta pump is proposed to predict the pressure head and blood flow of the intra-aorta pump. The model consists of a speed-controlled current source with an internal resistor and internal inductance. The speed current source represents the impeller that transforms kinetic energy to blood flow, the internal resistor represents the resistance character of the radial clearance, and the internal inductance mimics the inertia of the blood that passes through the radial clearance. The model can simulate overall situation of the pump and cardiovascular system from suction to pulmonary congestion. To identify the parameters and verify the model accuracy, benchtop experiments are conducted on an in vitro mock loop. The comparison results demonstrate that the model is able to predict the steady-state pressure head through the pump with an error <5% over the entire range of MAP specified for the intended use of the device.
Supported in part by the Beijing Nova Program (2006B12) and Scientific Research of Beijing Municipal Commission of Education (KM200710005033).
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